Estimation of relationships based on conditional analysis of identity-by-descent segments

Abstract

An example embodiment may involve obtaining first genetic sequence data of a first individual and second genetic sequence data of a second individual, wherein the first individual and the second individual are selected conditional on the first individual and the second individual sharing at least a threshold amount of identical-by-descent (IBD) segments of genetic sequence data; based on the first genetic sequence data and the second genetic sequence data, determining that the first individual and the second individual share N IBD segments of genetic sequence data, wherein the N IBD segments have respective lengths; and based on the N IBD segments of genetic sequence data and for a plurality of ancestral generations of the first individual and the second individual, estimating numbers of distinct common ancestors of the first individual and the second individual for each of the ancestral generations.

Description

Estimation of Relationships Based on Conditional Analysis of Identity-By-Descent SegmentsCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. provisional patent application no. 63 / 645,422, filed May 10, 2024, which is hereby incorporated by reference in its entirety. This application also claims priority to U.S. provisional patent application no. 63 / 688,163, filed August 28, 2024, which is hereby incorporated by reference in its entirety.GOVERNMENT LICENSE RIGHTS

[0002] This invention was made with government support under CONTRACT NO. R35GM133805 awarded by the National Institutes of Health. The government has certain rights in the invention.BACKGROUND

[0003] Identity-by-descent (IBD) is a term used to refer to matching segments of deoxyribonucleic acid (DNA) shared by two or more individuals, where these individuals inherited the segments from a common ancestor without recombination. IBD segments match when the alleles on a paternal or maternal chromosome are identical to one another above some pre-determined threshold. The length of IBD segments can vary based on the number of generations between the individuals and the ancestor (e.g., longer segments tend to be found when the common ancestor is more recent than when the common ancestor is not as recent).

[0004] The datasets of large genotyping biobanks contain many related individuals. Until now, it has been widely accepted that the most distant relationships that can be detected are around ten degrees, corresponding approximately to fifth cousins. In particular, relationship estimators are applied almost exclusively to putative relatives who have been identified because they share detectable IBD segments in their DNA. Due to limitations of current relationship estimators, relationship estimates are dramatically biased when applied to distant relationships, inferring all sufficiently distant relationships to be around ten degrees regardless of the depth of the true relationship.

[0005] A technique that is common in analyses of related individuals is the simulation of recombination events in pedigrees that relate the individuals to one another. Among other things, these simulations are used to obtain training and testing data for algorithm development, power analyses, the evaluation of logarithm of odds (LOD) scores in linkage studies, and theinference of relationships. Many methods have been developed for simulating meioses within a pedigree, but no method can sample meioses conditional on the event that individuals share at least one segment of IBD with one another. As a result, simulation studies that apply these methods to pedigrees that reflect information ascertained from IBD data produce biased results because conditional IBD sharing distributions are dramatically different from unconditional sharing distributions. This bias can be pronounced for pairs of individuals who are moderately or distantly-related (e.g., fourth cousins and higher).SUMMARY

[0006] A first example embodiment may involve: obtaining first genetic sequence data of a first individual and second genetic sequence data of a second individual, wherein the first individual and the second individual are selected conditional on them sharing at least a threshold amount of identical-by-descent (IBD) segments of genetic sequence data; based on the first genetic sequence data and the second genetic sequence data, determining that the first individual and the second individual share N IBD segments of genetic sequence data, wherein the N IBD segments have respective lengths; identifying a plurality of candidate shared ancestor sets for the first individual and the second individual, wherein each of the candidate shared ancestor sets includes a respective one or two possible shared ancestors with a respective first number of meioses between the first individual and the respective one or two possible shared ancestors and a respective second number of meioses between the second individual and the respective one or two possible shared ancestors; for each of the candidate shared ancestor sets, determining probabilities that the N IBD segments of the respective lengths would occur given that the first individual and the second individual share at least one IBD segment; and based on the probabilities, selecting one of the candidate shared ancestor sets as most-recent IBD-contributing shared ancestors of the first individual and the second individual.

[0007] A second example embodiment may involve: determining a relationship between a first individual and a second individual, wherein the relationship includes one or two possible shared ancestors of the first individual and the second individual, wherein a first number of meioses is between the first individual and the one or two possible shared ancestors, and wherein a second number of meioses is between the second individual and the one or two possible shared ancestors; determining that the first individual and the second individual share at least one identical -by-descent (IBD) segment of genetic sequence data; and probabilistically sampling a set of recombination breakpoints of each of the meioses sequentially, conditioned on any previously-sampled recombination breakpoints and conditioned on at least one IBDsegment being shared between the first individual and the second individual, wherein the set of recombination breakpoints is accepted when it meets an acceptance criterion.

[0008] A third example embodiment may involve: determining a pedigree including a set of relationships among a set of individuals, wherein each of the relationships has a respective one or two possible shared ancestors with a respective first number of meioses between a respective first individual and the respective one or two possible shared ancestors, and wherein each of the relationships also has a respective second number of meioses between a respective second individual and the respective one or two possible shared ancestors; determining which of the relationships are of individuals that share at least one IBD segment of genetic sequence data; and for each of the relationships, probabilistically sampling a set of recombination breakpoints of each of the meioses sequentially, conditioned on any previously- sampled recombination breakpoints and conditioned on at least one IBD segment being shared between individuals of the relationship, wherein the set of recombination breakpoints is accepted when it meets an acceptance criterion.

[0009] A fourth example embodiment may involve: obtaining first genetic sequence data of a first individual and second genetic sequence data of a second individual, wherein the first individual and the second individual are selected conditional on the first individual and the second individual sharing at least a threshold amount of identical-by-descent (IBD) segments of genetic sequence data; based on the first genetic sequence data and the second genetic sequence data, determining that the first individual and the second individual share N IBD segments of genetic sequence data, wherein the N IBD segments have respective lengths; and based on the N IBD segments of genetic sequence data and for a plurality of ancestral generations of the first individual and the second individual, estimating numbers of distinct common ancestors of the first individual and the second individual for each of the ancestral generations.

[0010] A fifth example embodiment may involve using the second and / or third example embodiment to generate training data for a machine learning algorithm, to generate testing data for a machine learning algorithm, to generate data from a distribution for the purpose of performing a power analysis to determine the power of a statistical test or analysis, or to empirically obtain a probability distribution on the number, lengths, or physical arrangement of IBD segments.

[0011] In a sixth example embodiment, an article of manufacture may include a non- transitory computer-readable medium, having stored thereon program instructions that, upon execution by a computing system, cause the computing system to perform operations inaccordance with any previous embodiment.

[0012] In a seventh example embodiment, a computing system may include at least one processor, as well as memory and program instructions. The program instructions may be stored in the memory, and upon execution by the at least one processor, cause the computing system to perform operations in accordance with any previous embodiment.

[0013] In an eighth example embodiment, a system may include various means for carrying out each of the operations of any previous embodiment.

[0014] These, as well as other embodiments, aspects, advantages, and alternatives, will become apparent to those of ordinary skill in the art by reading the following detailed description, with reference where appropriate to the accompanying drawings. Further, this summary and other descriptions and figures provided herein are intended to illustrate embodiments by way of example only and, as such, that numerous variations are possible. For instance, structural elements and process steps can be rearranged, combined, distributed, eliminated, or otherwise changed, while remaining within the scope of the embodiments as claimed.BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 illustrates a schematic drawing of a computing device, in accordance with example embodiments.

[0016] Figure 2 illustrates a schematic drawing of a server device cluster, in accordance with example embodiments.

[0017] Figure 3 A depicts an expected number of IBD-transmitting ancestors in each generation in the past, in accordance with example embodiments.

[0018] Figure 3B depicts an expected number of IBD-transmitting common ancestors who lived more than a number of generations in the past for various numbers of generations and various population sizes, in accordance with example embodiments.

[0019] Figure 3C depicts an inferred degree of relationship using unconditional and conditional estimators that assume a single common ancestor, in accordance with example embodiments.

[0020] Figure 3D depicts an inferred degree of relationship using a conditional estimator that allows for multiple common ancestors, in accordance with example embodiments.

[0021] Figure 4A depicts representations of a relationship usable by relationship estimators, in accordance with example embodiments.

[0022] Figure 4B depicts probability density functions of total IBD for several numbers of meioses, in accordance with example embodiments.

[0023] Figure 4C depicts ranges of relationship degrees, in accordance with example embodiments.

[0024] Figure 5 is a flow chart, in accordance with example embodiments.

[0025] Figure 6 is a flow chart, in accordance with example embodiments.

[0026] Figure 7 depicts a pedigree of individuals (circles) with meioses between them, in accordance with example embodiments.

[0027] Figure 8 depicts acceptance rules for a single meiosis rejection sampler, in accordance with example embodiments.

[0028] Figure 9 depicts the sampling of a meiosis, in accordance with example embodiments.

[0029] Figure 10 depicts a comparison of the simple rejection sampler with the Gibbs sampler, in accordance with example embodiments.

[0030] Figure 11 depicts the timing of the simple rejection sampler compared with timing of the Gibbs sampler, in accordance with example embodiments.

[0031] Figure 12 depicts a comparison of a rejection sampler and the Gibbs sampler for a pedigree with multiple constrained pairs, in accordance with example embodiments.

[0032] Figure 13 depicts a comparison of the unconditional and conditional sampling distributions, in accordance with example embodiments.

[0033] Figure 14 depicts a large pedigree on which meiosis can be simulated, in accordance with example embodiments.

[0034] Figure 15 is a flow chart, in accordance with example embodiments.

[0035] Figure 16 is a flow chart, in accordance with example embodiments.DETAILED DESCRIPTION

[0037] Example methods, devices, and systems are described herein. It should be understood that the words “example” and “exemplary” are used herein to mean “serving as an example, instance, or illustration.” Any embodiment or feature described herein as being an “example” or “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments or features unless stated as such. Thus, other embodiments can be utilized and other changes can be made without departing from the scope of the subj ect matter presented herein.

[0038] Accordingly, the example embodiments described herein are not meant to be limiting. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, separated, and designed in a wide variety of different configurations. For example, the separation of features into “client” and “server” components may occur in a number of ways.

[0039] Further, unless context suggests otherwise, the features illustrated in each of the figures may be used in combination with one another. Thus, the figures should be generally viewed as component aspects of one or more overall embodiments, with the understanding that not all illustrated features are necessary for each embodiment.

[0040] Additionally, any enumeration of elements, blocks, or steps in this specification or the claims is for purposes of clarity. Thus, such enumeration should not be interpreted to require or imply that these elements, blocks, or steps adhere to a particular arrangement or are carried out in a particular order.

[0041] The following U.S. patents and U.S. patent applications are incorporated by reference herein in their entirety: 8,463,554, 10,854,318, 11,031,101, 11,514,627, 2021 / 0020266, and 2022 / 0051751. However, such incorporation by reference is not an admission that the incorporated document is prior art with respect to the present disclosure.I. Definitions

[0042] The following terms are generally used as defined below. Nonetheless, given that many words have more than one meaning, the context in which such terms are used may modify their meanings accordingly.

[0043] The term “individual” refers to a single human person or organism for which genetic data is known or can be obtained, or for which genetic data cannot be obtained.

[0044] The terms “user” and “end user” refer to a person who uses a genetic computing platform to make queries and / or view results related to genetic data. This genetic data may be that of the user or end user, another individual, or a group.

[0045] The terms “genetic data” and “genetic datasets” refer to digitized representations of sequenced or partially sequenced genomes of individuals.

[0046] The term “genetic computing platform” refers to one or more computing devices or systems (e.g., processing, storage, and / or communication hardware and / or software) that are configured to carry out the embodiments herein.

[0047] The term “allele” refers to a variant form of a gene at a specific locus on a chromosome, which contributes to differences in inherited traits among individuals. Each human inherits two alleles per gene, one from each parent.

[0048] The term “haplotype” refers to a set of genomic polymorphisms that are physically located near one another on a chromosome and tend to be inherited together.II. Example Computing Devices and Cloud-Based Computing Environments

[0049] Figure 1 is a simplified block diagram exemplifying a computing device 100, illustrating some of the components that could be included in a computing device arranged to operate in accordance with the embodiments herein. Computing device 100 could be a client device (e.g., a device actively operated by a user), a server device (e.g., a device that provides computational services to client devices), or some other type of computational platform. Some server devices may operate as client devices from time to time in order to perform particular operations, and some client devices may incorporate server features.

[0050] In this example, computing device 100 includes processor 102, memory 104, network interface 106, and input / output unit 108, all of which may be coupled by system bus 110 or a similar mechanism. In some embodiments, computing device 100 may include other components and / or peripheral devices (e g., detachable storage, printers, and so on).

[0051] Processor 102 may be one or more of any type of computer processing element, such as a central processing unit (CPU), a co-processor (e.g., a mathematics, graphics, or encryption co-processor), a digital signal processor (DSP), a network processor, and / or a form of integrated circuit or controller that performs processor operations. In some cases, processor 102 may be one or more single-core processors. In other cases, processor 102 may be one or more multi-core processors with multiple independent processing units. Processor 102 may also include register memory for temporarily storing instructions being executed and related data, as well as cache memory for temporarily storing recently-used instructions and data.

[0052] Memory 104 may be any form of computer-usable memory, including but not limited to random access memory (RAM), read-only memory (ROM), and non-volatile memory (e.g., flash memory, hard disk drives, solid state drives, compact discs (CDs), digital video discs (DVDs), and / or tape storage). Thus, memory 104 represents both main memoryunits, as well as long-term storage. Other types of memory may include biological memory.

[0053] Memory 104 may store program instructions and / or data on which program instructions may operate. By way of example, memory 104 may store these program instructions on a non-transitory, computer-readable medium, such that the instructions are executable by processor 102 to carry out any of the methods, processes, or operations disclosed in this specification or the accompanying drawings.

[0054] As shown in Figure 1, memory 104 may include firmware 104A, kernel 104B, and / or applications 104C. Firmware 104A may be program code used to boot or otherwise initiate some or all of computing device 100. Kernel 104B may be an operating system, including modules for memory management, scheduling and management of processes, input / output, and communication. Kernel 104B may also include device drivers that allow the operating system to communicate with the hardware modules (e g., memory units, networking interfaces, ports, and buses) of computing device 100. Applications 104C may be one or more user-space software programs, such as web browsers or email clients, as well as any software libraries used by these programs. Memory 104 may also store data used by these and other programs and applications.

[0055] Network interface 106 may take the form of one or more wireline interfaces, such as Ethernet (e.g., Fast Ethernet, Gigabit Ethernet, and so on). Network interface 106 may also support communication over one or more non-Ethernet media, such as coaxial cables or power lines, or over wide-area media, such as Synchronous Optical Networking (SONET) or digital subscriber line (DSL) technologies. Network interface 106 may additionally take the form of one or more wireless interfaces, such as IEEE 802.11 (Wifi), BLUETOOTH®, global positioning system (GPS), or a wide-area wireless interface. However, other forms of physical layer interfaces and other types of standard or proprietary communication protocols may be used over network interface 106. Furthermore, network interface 106 may comprise multiple physical interfaces. For instance, some embodiments of computing device 100 may include Ethernet, BLUETOOTH®, and Wifi interfaces.

[0056] Input / output unit 108 may facilitate user and peripheral device interaction with computing device 100. Input / output unit 108 may include one or more types of input devices, such as a keyboard, a mouse, a touch screen, and so on. Similarly, input / output unit 108 may include one or more types of output devices, such as a screen, monitor, printer, and / or one or more light emitting diodes (LEDs). Additionally or alternatively, computing device 100 may communicate with other devices using a universal serial bus (USB) or high-definition multimedia interface (HDMI) port interface, for example.

[0057] One or more computing devices like computing device 100 may be deployed to support the embodiments herein. The exact physical location, connectivity, and configuration of these computing devices may be unknown and / or unimportant to client devices. Accordingly, the computing devices may be referred to as “cloud-based” devices that may be housed at various remote data center locations.

[0058] Figure 2 depicts a cloud-based server cluster 200 in accordance with example embodiments. In Figure 2, operations of a computing device (e.g., computing device 100) may be distributed between server devices 202, data storage 204, and routers 206, all of which may be connected by local cluster network 208 The number of server devices 202, data storages 204, and routers 206 in server cluster 200 may depend on the computing task(s) and / or applications assigned to server cluster 200.

[0059] For example, server devices 202 can be configured to perform various computing tasks of computing device 100. Thus, computing tasks can be distributed among one or more of server devices 202. To the extent that these computing tasks can be performed in parallel, such a distribution of tasks may reduce the total time to complete these tasks and return a result. For purposes of simplicity, both server cluster 200 and individual server devices 202 may be referred to as a “server device.” This nomenclature should be understood to imply that one or more distinct server devices, data storage devices, and cluster routers may be involved in server device operations.

[0060] Data storage 204 may be data storage arrays that include drive array controllers configured to manage read and write access to groups of hard disk drives and / or solid state drives. The drive array controllers, alone or in conjunction with server devices 202, may also be configured to manage backup or redundant copies of the data stored in data storage 204 to protect against drive failures or other types of failures that prevent one or more of server devices 202 from accessing units of data storage 204. Other types of memory aside from drives may be used.

[0061] Routers 206 may include networking equipment configured to provide internal and external communications for server cluster 200. For example, routers 206 may include one or more packet-switching and / or routing devices (including switches and / or gateways) configured to provide (i) network communications between server devices 202 and data storage 204 via local cluster network 208, and / or (ii) network communications between server cluster 200 and other devices via communication link 210 to network 212.

[0062] Additionally, the configuration of routers 206 can be based at least in part on the data communication requirements of server devices 202 and data storage 204, the latencyand throughput of the local cluster network 208, the latency, throughput, and cost of communication link 210, and / or other factors that may contribute to the cost, speed, faulttolerance, resiliency, efficiency, and / or other design goals of the system architecture.

[0063] As a possible example, data storage 204 may include any form of database, such as a structured query language (SQL) database. Various types of data structures may store the information in such a database, including but not limited to tables, arrays, lists, trees, and tuples. Furthermore, any databases in data storage 204 may be monolithic or distributed across multiple physical devices.

[0064] Server devices 202 may be configured to transmit data to and receive data from data storage 204. This transmission and retrieval may take the form of SQL queries or other types of database queries, and the output of such queries, respectively. Additional text, images, video, and / or audio may be included as well. Furthermore, server devices 202 may organize the received data into web page or web application representations. Such a representation may take the form of a markup language, such as HTML, the extensible Markup Language (XML), JavaScript Object Notation (JSON), or some other standardized or proprietary format. Moreover, server devices 202 may have the capability of executing various types of computerized scripting languages, such as but not limited to Perl, Python, PHP Hypertext Preprocessor (PHP), Active Server Pages (ASP), JAVASCRIPT®, and so on. Computer program code written in these languages may facilitate the providing of web pages to client devices, as well as client device interaction with the web pages. Alternatively or additionally, JAVA® may be used to facilitate generation of web pages and / or to provide web application functionality.III. Improved Relationship Estimators

[0065] Geneticists, scientists, medical professionals, genealogists, and individuals may have various reasons to want to be able to cluster individuals into genetic groups For instance, such groupings may be helpful in understanding genetic relationships, migratory patterns, language evolution, and / or predispositions for certain diseases or health conditions. Other benefits, such as gaining further insight into one’s ancestry may be desirable or beneficial.

[0066] As noted above, relationship estimators are most commonly applied to putative relatives who have been identified because they share detectable segments of DNA that are identical by descent (IBD). These segments are tracts of DNA that two or more individuals inherit from a common ancestor without alteration through recombination. IBD segments are markers for determining genetic relationships and ancestry because their presence indicates a direct genealogical link between the individuals sharing them. In population genetics andgenealogical studies, the analysis of IBD segments is used for assessing the degree of relatedness among individuals, mapping genetic traits, and understanding the patterns of inheritance and migration within and between populations. Additionally, the study of IBD segments is employed in medical genetics to identify inherited disease risks, as diseases can be linked to specific genes that have been passed down through generations. This approach is particularly useful in the study of complex genetic diseases where multiple genes may contribute to disease susceptibility.

[0067] In spite of the prevalent use of IBD analyses in genetics-related research and medicine, no existing relationship estimator conditions on the event that two individuals share IBD. As a result, the relationship estimates obtained using existing estimators are dramatically biased when applied to distant relationships, inferring all sufficiently distant relationships to be around ten degrees regardless of the depth of the true relationship. This significantly limits the ability of IBD-based estimators to accurately detect distant relationships between living relatives for purposes of relationship or pedigree determination.

[0068] Moreover, existing relationship estimators are based on an incorrect model of relationship in which each individual is related to each of their genealogical relatives through only one common ancestor or pair of common ancestors. In reality, each pair of individuals likely shares many common ancestors in many different generations in the past.

[0069] The embodiments herein provide a corrected likelihood estimator that conditions on the event that at least one IBD segment is observed between a pair of putative relatives and that largely eliminates the bias in estimates of distant pairwise relationships. With these improvements, the deepest relationship that can be detected is closer to two hundred degrees, rather than ten. This dramatic increase in the range of relationship estimators allows inference of relationships whose common ancestors lived before historical events such as European settlement of the Americas and the Transatlantic Slave Trade, for example.

[0070] The embodiments herein also present a new model for relationship estimation in which each pair of putative relatives is related through many common ancestors in the past. This model makes it possible to infer the complex relationships that truly exist among people.

[0071] A genetic relationship inference method is an algorithm that takes as input the genotyped or sequenced DNA of a putative pair of relatives, along with their ages, sexes and potentially other information, and returns an estimate of their relationship. These algorithms are commonly applied in the context of genetic testing in order to identify relatives and infer genealogies, and they can be applied in the context of medical genetic studies to identify and leverage or control for cryptic relatedness.

[0072] Current relationship inference methods rely on probability distributions that describe how much IBD is observed between a pair of individuals as a function of their relationship. Common statistics include the total length of observed IBD in centimorgans or other quantities such as the kinship coefficient, and / or the number of observed IBD segments and their lengths.

[0073] Regardless of the approach, all existing methods rely implicitly or explicitly on distributions of IBD statistics that were obtained without conditioning on the event that IBD was observed between the two putative relatives. For example, some likelihood methods rely on the probability distribution of one or more observed IBD statistics. Estimators commonly used in genetic testing rely on empirical versions of these probability distributions that can be obtained using simulations. Other methods rely on analytically-derived bounds that delineate regions of “IBD space” in which the observed values of IBD statistics are most consistent with different relationships.

[0074] All of these distributions are unconditional on IBD sharing. While some derive a conditional version of the probability distribution of observed segment lengths conditional on the event that two putative relatives are ascertained because they share IBD at a particular locus, this distribution is not the same as the distribution conditional on observing at least one segment of IBD anywhere in the genome.

[0075] Failure to condition on observing at least one IBD segment is appropriate in scenarios in which it is known or suspected that two individuals are related; for example, to verify a self-reported relationship that was identified based on previous genealogical knowledge. However, in most contexts in which relationship inference is applied, pairs of putative relatives are identified by first detecting shared IBD. In this context, it is inappropriate to apply estimators that do not condition on the event that IBD is observed.

[0076] Failure to condition on the observation of IBD has relatively little effect when a relationship is close because the probability is high that close relatives share IBD. However, when relationships are distant, failure to condition on the event that IBD is observed has profound consequences resulting in dramatically biased relationship estimates.

[0077] The embodiments herein derive the probability distribution of the observed number of IBD segments and their lengths as a function of the genealogical relationship that gave rise to the segments, conditional on the event that at least one segment of IBD was observed. The resulting corrected estimator no longer has the profound bias observed in the unconditional estimator and it allows relationship estimates that stretch into the distant past.

[0078] An additional embodiment herein presents an improved relationship estimatorthat allows two relatives to be related through multiple common ancestors. This embodiment derives the probability of observing n IBD segments with total length L, conditional on multiple shared ancestors and on the event that each shared ancestor contributed at least one segment of observed IBD. The model of relatedness underlying this multiple-ancestor estimator is more realistic, especially for distant relationships.A. Observations

[0079] Before investigating the bias in existing relationship estimators, it is informative to consider how often distant IBD-sharing relationships can be expected to appear in practice. If each pair of relatives has a single common ancestor or a single mating pair of common ancestors, then the probability of sharing IBD with a distant relative is indeed very small. For example, sixth cousins typically share just one IBD segment with one another, while 8th cousins and beyond (individuals who share a pair of common ancestors nine generations or more in the past) are exceedingly unlikely to share any detectable IBD segments at all - the probability that 8th cousins share at least one segment is less than 1%.

[0080] Although it is very unlikely for two distant relatives to share detectable IBD through a particular common ancestor, the number of common ancestors shared between any two individuals at each generation in the past is large. Moreover, due to pedigree collapse (the fact that the number of genealogical ancestors of an individual increases by a factor of two at each generation in the past and quickly exceeds the number of available individuals in the population, implying that each living ancestor many generations in the past must have corresponded to multiple genealogical ancestors), each modern individual can have multiple semi-independent lineages back to each of their common ancestors. Thus, the chances of observing a very distant IBD-sharing relationship are actually quite high.

[0081] The embodiments herein may involve obtaining an expression for the expected number of distinct IBD-transmitting common ancestors shared by two individuals at each generation in the past. This distribution provides the fraction of an individual’s IBD-sharing relatives whose IBD arose from a common ancestor living in each past generation.

[0082] Figure 3 A shows the expected number of IBD-transmitting common ancestors at each generation in the past along with empirical expectations obtained from simulations. Figure 3A demonstrates that the analytical expectations are close to the empirical expected number of IBD-transmitting ancestors for several population sizes for which simulation is computationally efficient. The similarity breaks down for large generation times. For large generation times, the analytical formulas underestimate the empirical expected number of ancestors and are therefore conservative. This is clearly observable from the log scale plot ofFigure 3 A.

[0083] Figure 3B shows the expected number of IBD-transmitting common ancestors who lived at least g generations in the past for various values of g and for effective population sizes N that are reasonable for human populations (e.g., 1,000 Si N100,000). For all population sizes, approximately 60% of all IBD-sharing relationships are through an ancestor who lived more than ten generations in the past (approximately 20 degrees). The expectations in Figure 3B were computed for several different effective population sizes ranging from very small sizes (IV = 100) to reasonable sizes (N « 10,000 or even N 100,000) for human populations within the past.

[0084] Thus, IBD-sharing relationships greater than 20 degrees occur quite frequently. As a result, relationship estimators that are capable of inferring these distant relationships are possible. Herein, the term “degree” is approximately the number of meioses between two individuals (e.g., the number of meioses between the first individual and their most common ancestor plus the number of meioses between the second individual and their most common ancestor). A more precise definition is given below.

[0085] To understand the bias that results when unconditional relationship estimators are applied to relatives ascertained on the basis of IBD sharing, IBD was simulated between relatives of various degrees, conditional on the event that they shared at least one detectable segment of IBD with one another. This sampling scheme reflects what is expected in most real data applications involving relative detection in genetic testing or biobank data.

[0086] A count of 10,000 relative pairs were sampled uniformly with respect to the degree of the relationship between one and eighty-one degrees. For each simulation replicate, the degree of the relationship between the pair of individuals was inferred by maximizing the unconditional likelihood and separately by maximizing the conditional likelihood derived below.

[0087] Figure 3C depicts this inferred degree using unconditional and conditional estimators. Panel A of Figure 3C shows the inferred degree using the unconditional likelihood compared with the true degree for 10,000 simulated relationships between 1 and 81 degrees. Panel B of Figure 3C shows the inferred degree using the conditional likelihood (based on Equation 8, below).

[0088] From Panel A of Figure 3C, it can be seen that the unconditional relationship estimates are fairly accurate for close relationships up to approximately ten degrees (approximately fourth cousins), but they begin to diverge sharply from the true degree forrelationships beyond ten degrees. Moreover, as the degree of the true relationship increases, the estimated relationships become increasingly tightly grouped around ten degrees.

[0089] In contrast to the unconditional estimates, the conditional estimates shown in Panel B of Figure 3C have considerably less bias. For these estimates, the mean inferred degree tracks reasonably well with the true degree (the diagonal line). The trade-off for reduced bias is increased variance, which can be seen in the range of inferred values shown in the boxplot. This increased variance is due to the fact that there is typically only one segment shared between distant relatives and all information about the degree of the relationship is contained in the length of the segment. Segment lengths for distant relationships do not carry much information about the true relationship degree, as the segment length distributions for different degrees overlap considerably.

[0090] Although the estimates in Panel B of Figure 3C are much less biased than the estimates in Panel A, both estimators assume a single common ancestor. Figure 3D shows estimates made under an estimator that allows each related pair to have multiple common ancestors. This estimator computes the most likely set of common ancestors at multiple generations in the past that gave rise to the observed IBD. It then finds the most recent of these most likely ancestors. From Figure 3D, it can be seen that this estimator also tracks reasonably well with the true degree of relationship for simulated pairs of relatives who have just one common ancestor.B. Improved Maximum Likelihood Relationship Estimator

[0091] Before deriving a new conditional relationship estimator, it is useful to clarify what is meant by inferring “the relationship” between a pair of individuals. Existing relationship estimators are derived under the conceptual model shown in Panel A of Figure 4A. In this model, two individuals, i and j (represented as large circles at the bottom of Panel A of Figure 4A) are related through a single recent common ancestor, a, or through a mating pair of recent common ancestors (a15a2)- Individuals i and j may have other very distant ancestors (smaller circles at the top of Panel A of Figure 4A) that give rise to occasional small segments of “background IBD,” but in this model, background IBD reflects very distant relationships that are outside of scope of relationship inference. Instead, background IBD segments produce noise that confounds the signal of the close relationship (the curve) that is the focus of inference.

[0092] The assumption that two people are connected through a single close relationship may be true when attention is restricted to the very recent past (perhaps within the most recent five to ten generations), but this assumption quickly breaks down as the degree of the relationship increases. As discussed, because of pedigree collapse, each pair of individualstypically has many genealogical ancestors in each generation in the past, only some of whom contributed IBD that is observed between the pair.

[0093] Panel B of Figure 4A shows a model that more accurately describes genealogical relatedness at distant timescales. In this model, each pair of individuals shares many common ancestors in each generation in the past (smaller circles throughout Panel B of Figure 4A). Some of these ancestors contribute observable IBD to the pair and some do not.

[0094] As discussed below, the number of common genealogical ancestors is large and each individual has many lineages connecting them to each ancestor due to pedigree collapse. Therefore, for distant relationships, it is more appropriate to conceptualize the relationship inference problem in the form shown in Panel B of Figure 4A.

[0095] The goal of relationship inference under the model in Panel B of Figure 4A is not just to infer “the relationship” between i and j since there is not just one relationship. Instead, possible goals include inferring any one of several quantities of interest such as: (1) the most recent genealogical relationship, (2) the most recent genealogical relationship that resulted in observable shared IBD, (3) the number of genealogical ancestors at each generation in the past, (4) the number of ancestors at each generation in the past that contributed observable shared IBD, or some other suitable quantity that reflects the fact that individuals share many common ancestors through many different relationships.

[0096] In the embodiments herein, inheritance is framed under the model in Panel B of Figure 4A and statistic 2 is inferred (the genealogically closest relationship that contributed observable IBD). This quantity is close to the idea of inferring “the relationship” between a pair of IBD-sharing individuals.1. The Conditional Likelihood

[0097] Before deriving a relationship estimator that considers multiple ancestors, it is useful to see how conditioning on the observation of at least one IBD segment affects existing relationship estimators derived under the model shown in panel A of Figure 4A. The distribution of the number and lengths of observed IBD segments, conditional on observing at least one segment, can be derived by the following method. Let R denote the relationship between individuals, i and j, where R = (u, v, a) indicates that i and j are related through a G {1, 2} common ancestors) with u meioses separating i from the ancestor(s) and v meioses separating j from the ancestor(s). The total number of meioses is m — u + v and the degree is d = m — a + 1. Thus, relatives with a degree of 1 share approximately 50% of their genes, relatives with a degree of 2 share approximately 25% of their genes, relatives witha degree of 3 share approximately 12.5% of their genes, and so on.

[0098] Let n denote the number of IBD segments shared between relatives i and j . Some of these segments come from the most recent common ancestor who contributed detectable IBD to i and j, giving rise to the relationship to infer. Other segments come from other ancestors. Let nddenote the number of segments that came from the common ancestor of interest and let nbdenote the number of segments that came from other ancestors.

[0099] Letdenote the lengths of the n = nd+ nbIBD segments observed between i and j in units of centimorgans. Let O be the event that i and j are observed to share at least one segment of IBD. A goal is to compute the probability P( i, . . .,-£n\O; m, a) of the observed IBD, conditional on the event, O, that at least one IBD segment is observed. This probability is a function of the number, a, of most-recent common genealogical ancestors and the number, m, of meioses that separate i and j.

[0100] To simplify, the assumption can be made that the ndsegments coming from the most-recent IBD-contributing common ancestor(s) are the longest segments. This assumption allows avoiding having to condition on the subset of IBD segments that arose from this ancestor, which allows avoiding a summation over all subsets of segments that could have come from the common ancestor. Given this simplifying assumption, the distribution of segment lengths is:

[0101] wheredenotes the length of the ith longest segment and P^C) denotes the probability distribution of background IBD segment lengths. The terms in Equation (1) can beobtained as follows:

[0102] where Aa mis the inverse of the expected length of an IBD segment between two people separated by m meioses and T is the minimum observable segment length in centimorgans. The segment lengths are modeled as independent, which is likely to be an accurate approximation when m is moderate to large. The approximation Aa m« m / 100 is used, in which case the right-hand side of Equation (2) does not depend on a.

[0103] Similarly, the termcan be modeled as the product of n — ndindependent exponential distributions:(3)is the inverse length of a background IBD segment. The value of A{can be found empirically by assuming that most pairs of individuals in a large database are only distantly-related and averaging the lengths of IBD segments shared between many randomly sampled pairs.

[0105] The distribution P(nb= i) of the number, nb, of background segments can be found empirically as well. In particular, it is reasonable to model nbas a Poisson random variable:

[0106] with mean T)bequal to the average number of IBD segments observed between a randomly-sampled pair of individuals in a dataset.

[0107] The number of observed segments from the genealogically most recent IBD- contributing common ancestors) can be modeled Poisson, with mean r)a m T:

[0108] where, for moderate-to-large m, T]a>m>Tcan be approximated as

[0109] where r is the expected number of recombination events per meiosis in the genome, c is the number of chromosomes, and T is the minimum observable segment length. In humans, r « 35 and c = 22.

[0110] Finally, the probability of observing any segments at all is:

[0111] which comes from the fact that that ndand nbare each Poisson, so their sum isPoisson with mean equal to the sum of the individual means. Equation (7) is one minus the probability that the sum nd+ nbis zero.

[0112] All together, Equation (1) becomes:Equation (8) suggests a simple derivation, as the conditional distribution is obtained from the unconditional expression by ignoring the mass at zero and renormalizing the mass above zero.

[0114] Further, if there is no background IBD, the distribution of the total length of IBD conditional on observing at least one IBD segment can be found as follows. The total length of shared IBD £, conditional on the number n of observed segments and on the event O is the sum of n exponential random variables, each with mean 1 / Aa m. L follows a gamma distribution with parameters n and l / 2a m. The joint distribution of the total number n of segments and the total amount L of IBD is then:

[0115] where rja m Tis given above in Equation (6). The distribution of the total length L of IBD alone can be obtained. To do this, the number of observed IBD segments is conditioned on as follows:

[0116] For the case in which the minimum segment length is T = 0, this equation has a closed form, which can be obtained by rearranging the terms to resemble the summation in a modified Bessel function:

[0117] where Ia(z) is the modified Bessel / function of the first kind. The summation in the penultimate equality is the expression for the modified Bessel function of the first kind with a = 1. Figure 4B shows the distribution of the total length for the case of a = 1 common ancestors and several values of m Figure 4B is the analytical and conditional version of the empirically-obtained unconditional distributions.

[0118] Note that the only aspect of fi\o(L\0 that corresponds to conditioning on the event 0 is the factor 1 — e-77a-m-Tin the denominator. Thus, to obtain the unconditional distribution of the total length, this factor can be removed:

[0119] This equation for fL(L) provides the analytical version of the empirical distribution that is used by genetic testing companies for relationship inference. Genetic testing companies typically obtain these distributions empirically using simulations. They simulate IBD for two related individuals at increasingly large degrees of relationship and record the empirical distributions of the total lengths of IBD. Because there is noise in empirical simulations, the distributions obtained by these companies are very noisy. The analytical distribution has zero noise.

[0120] From the above, thresholds that are used for relationship inference can be analytically obtained and extended to the case of conditional likelihoods. Using distributionsobtained with empirical simulations, the range of possible total IBD lengths L E [0, oo) can be partitioned into regionssuch that the most likely degree of relationship in the range (Ld, Ld-1] is d.

[0121] The range (Ld- ^d-i] depends only on the degree d = m — a + 1, rather than explicitly on a and m. For distant relationships, this dependence on d alone is reasonable because a relationship of degree d with two common ancestors a = 2 has a very similar pattern of shared IBD compared with a relationship of degree d and one common ancestor a = 1. Although age information can help to determine the number of ancestors a for close relationships, for very distant relationships, there is almost no available information to determine whether the number of ancestors is 2 or 1. Hence, the degree d is often the most relevant quantity to infer for distant relatives.

[0122] Using these equations, ranges can be obtained analytically. Such an analytically derived estimator is arguably better than the empirically-derived estimators that are currently used because there is no simulation noise to contend with when computing the boundary points Ld. To find the boundary point Ldanalytically, a can be set to either 1 or 2 to make the equations a function of the degree d. Fo| becomes[0123. To find the boundary point Ldone can find the value of L such that Solving this equationyields the ranges shown in Figure 4C.

[0124] The ranges in Figure 4C explains why relationships are never inferred to be more distant than around tenth degree using the unconditional estimators, even though plenty of more-distant relationships still generate detectable IBD, even under unconditional simulations. The reason is that the point L10is at zero. This means that the likelihood surface for the case m = 10 is always greater than the likelihood surface for m > 10 in the range (0, oo) , i.e., for values of L greater than 0. This occurs because the distribution for the unconditional case has a point mass at L = 0, corresponding to the case in which zero IBD segments are observed. As a result, for nonzero detected IBD less than 3.39 cM, the most likely relationship in the unconditional case is always 10 degrees. In contrast, the conditional estimator has resolution for much more distant relationships.2. The Prior Distribution of the Number of Generations to the Most Recent IBD-Segment Contributing Common Ancestor

[0125] Equation (8) can be used to obtain a maximum likelihood estimate of therelationship, given the observed IBD segments. However, it may also be of interest to perform inference in a Bayesian context because the relative probability of observing a segmentcontributing common ancestor at each generation in the past can be informative.

[0126] To obtain the expected number of detectable-IBD-contributing common ancestors at each generation in the past, a more fundamental quantity can be first obtained - the expected numbe of detectable transmitted IBD segments arising in generation g. Toobtain note that all of the DNA in a person living in the present day came from their setof ancestors living in generation g in the past. Moreover, g generations in the past, each copy of a present-day individual’s genome existed in approximatel iny semi-independenthaplotype chunks that would ultimately combine over the generations to produce the linear genome of the present-day person. The expected number E[Hg] of haplotypes chunks is:

[0127] Equation (9) comes from the fact that a genome has, on average, rg + c — 1 breaks introduced into it over g generations including the obligatory breaks at chromosome endpoints, yieldinpieces on average.

[0128] Considering the overlapping chunks of two linear genomes g generations in the past, the rc + g chunks from one copy of i’s genome will overlap the rc + g chunks from a copy of j’s genome in an average of 2(rg + c) overlapping regions. The length of one of these overlapping chunks between two people found anywhere in the genome is exponentially distributed with mean 100 / 2$. Therefore, the probability that a shared segment anywhere in the genome is longer than r cM is thenunder the coalescent model, the probability that a shared segment between two individuals coalesces g generations in the past in a population with N diploid individuals is e~g 2N. Putting all of this together yields:Here, the factor of 4 comes from the fact that individuals i and j each have two copies of the linear genome that can coalesce with one another.

[0129] Equation (10) can be used to find the expected number of distinct detectable- IBD-transmitting ancestors in generation g . If i and j share s segments through common ancestors living g generations ago, then the probability they came from exactly k distinct common ancestors can be obtained as follows. Let c be the number of common ancestors. Thenumber of ways of placing s indistinguishable segments into c individuals isnumber of ways of choosing k specific individuals is Q). Finally, the number of ways of placing s indistinguishable segments into k specific individuals so that each individual has at least one segment is J. Note that each of the s shared segments must have come from acommon ancestor. So, all together, the probability that there are k distinct ancestors in generation g, given that there are s shared segments is:

[0130] Thus, the expected number of common ancestors, given that there are s segments is:

[0131] Modeling the number of shared segments Sgarising in generation g as Poisson distributed with mean E[5fl], the expected number of distinct common ancestors Kgis:

[0132] Where gg— E[S^] is given by Equation (10).3. Multi-Ancestor Likelihoods

[0133] The estimator in Equation (8) still implicitly uses the model of relatedness shown in Figure 4A in which each pair of relatives has exactly one common ancestor or pair of common ancestors. A relationship estimator that explicitly uses the multi-ancestor model shown in Panel B of Figure 4A can be derived by assuming that each pair of individuals has many common ancestors and that there is no such thing as background IBD. Such an estimator can more accurately reflect the nature of true relatedness in a population.

[0134] As noted, several statistics are more appropriate for capturing the complexity of true relation-ships, such as 1) the time to the most recent genealogical ancestor, 2) the time to the most recent IBD-contributing common ancestor, 3) the number of genealogical ancestorsin each generation in the past, and 4) the number of detectable-IBD-transmitting common ancestors at each generation in the past. Statistics 2 and 4 are perhaps easier to infer because they pertain to ancestors who contributed observed IBD. The embodiments herein focus on these statistics to avoid the extra layer of complexity involved in inferring ancestors who left no genetic traces of themselves in the putative relative pair.

[0135] Suppose n segments are observed with a total of L cM shared between a pair of individuals. An estimate of the number of ancestors who transmitted these segments and the generations in which they lived can be obtained from the probability of the observed IBD, given the full, complex ancestral relationship of the two individuals.

[0136] Suppose the two people share a total of K common ancestors who lived in generations g G G in the past for some set of generations G . Let kgbe the number of ancestors in generation g so th be the degree separating a pair of individuals witha common ancestor in generation g. For example if the two people are contemporaneous, then dg= 2g, but they need not be contemporaneous.

[0137] The joint distribution of n and L can be found by summing over the number of segments attributable to ancestors in each generation. Let ngbe the number of segments inherited from the ancestors in generation g . The generation from which a segment was inherited specifies the expected length 9gof the segment. Since the segment lengths are exponentially distributed, the expected total length Lgof the ngsegments arising in generation g follows a Gamma distribution with shape parameter ngand scale parameter 6g.

[0138] The total observed lengt can be approximated as the sum of |G|independent but not identically distributed gamma-distributed random variables. L can be approximated by a single gamma-distributed random variable whose mean and variance match the mean and variance of the sumg ecLg. Since the Lgare independent, the mean and variance of L are:

[0139] Here, the summands come from the fact that the parameters a and ? of a gamma-distributed random variable X satisfy £[X] = a and var(X) = aft2. It follows that the parameters aLand of L, given {ng)geGsatisfy:

[0140] Here, the mean segment length Qgfrom an ancestor in generation g is 100 / dgand dgis the degree of the relationship between the two relatives i and j that passes through an ancestor in generation g.

[0141] Given that n = {ng]geGsegments are transmitted from each generation g 6 G and that the degrees of the induced relationships are {dg}geG, the observed number and total length of IBD has the distribution:Here, aLand are given by Equations (14) and (15).

[0142] To obtain the distribution of n and L it is remains to find the probability of n = {ng}geG conditional on and the fact that each ancestor in k contributed at leastone segment longer than T. Since there are E[Kg] ancestors on average in generation g and an expected number of IBD segments, that means that each ancestor contributes E[Sfl] / E[cfl] segments on average, where cgis the number of common genealogical ancestors in generation g . If there are kgancestors in generation g then the expected number of contributed segments, beyond the kgobligatorily contributed segments iwhere cgis the expected number of distinct common ancestors in generation g.

[0143] Putting this all together:

[0144] Here, the final equality comes from modeling the ngsegments from the kgancestors in generation g as Poisson distributed with mean yg, conditional on the event thatng> kg. Here, ygis given by:

[0145] This is kgtimes the expected number of lineages transmitted to two offspring by a single common ancestors living g generations in the past.

[0146] Equation (18) is the expected number of shared segments, assuming that each ancestor is connected to each descendant by just one lineage. To account for the fact that each ancestor is connected to each descendant by multiple lineages, vgcan be expressed as:

[0147] Here, cgis the number of distinct detectable-IBD-transmitting common ancestors in generation g. Thus, times the expected number of segments perdetectable-IBD-transmitting common ancestor, or the total number of IBD segments transmitted from ancestors in generation g.

[0148] The quantity can be derived as follows. Let agdenote the number ofdistinct ancestors in generation g. Although each person has 23total ancestors in generation g, there are N total individuals in the population meaning that some of these ancestors are the same person whenever A < 29. Assuming a randomly-mating population in which the two parents of each individual in generation g — 1 are chosen uniformly at random from among the N individuals in the population in generation g, agcan be modeled as the number of distinct draws from a population of N distinct individuals when 2afl-1samples are taken with replacement.

[0149] To develop a recurrence relation for cgand ag, let 6gdenote the number of distinct ancestors of individual i in generation g that are not common ancestors of individual denote the number of distinct common ancestors shared between individuals i and jin generation g. Finally, let 8gdenote the number of distinct ancestors of the 6g- }individualancestors of i in generatio 1 before considering mergers with ancestors of j, and letdenote the number of distinct ancestors of the c5-xcommon ancestors before considering mergers among individuals. Thus Sgis the number of distinct objects when drawinsamples with replacement from a population of N individuals and cgis the number of distinct individuals selected when drawing 2cg-1objects with replacement from a population of N individuals.

[0150] The quantity agcan be obtained from the quantity aq-by noting that the probability that any given one of the N individuals in generation g is a parent of one of the individuals in generation g — 1 with probabili s probabilitycomes from the fact that the individual is selected in a single draw from the population with probability 1 / N. Thus the probability that they are not selected in any of the 2ag-±draws is (1 — l / lV)2a5-i . it follows that agis binary with probability 1 — (1 —and ] = 7V[1 — (1 — 1 /

[0151] When the population size is large, the quantities 8g, 8g, cgand cgbehave quasi- deterministically and approximately follow their expected values. Thus, in generation g, the 8gdistinct non-common ancestors of i overlap with one of the cgdistinct common ancestors at rate 6gcg / N2and they overlap with one of the 6gdistinct ancestors of individual j at rate 62g / N2. Therefore, the expected number of distinct common ancestors at generation g is:

[0153] Repeatedly applying this recursion gives an approximation of the number of distinct common ancestors at each generation in the past.C. Further Aspects

[0154] The embodiments herein establish that existing relationship estimators (e g., those that do not condition on the event that IBD is observed between a pair of relatives) produce profoundly biased estimates when the degree of relationship is greater than ten. It has also been demonstrated that IBD-sharing relationships of degree greater than ten are ubiquitous, amounting to a large percentage of all relationships that are detectable in the population.

[0155] Given the profound nature of the bias in existing relationship estimators, it is natural to consider why this problem has remained unobserved for so long. One reason is that the bias in relationship estimators tends to force estimates of distant relationships downwards towards ten degrees or so, and it has generally been assumed that the most distant detectablerelationships are not much greater than that. In particular, because two distantly-related individuals are very unlikely to share IBD at all, it was assumed that relationships beyond fifteen degrees (8th cousins) were generally undetectable. The failure of existing estimators to detect such distant relationships was, therefore, taken as evidence of a fundamental truth about relatedness, rather than raising suspicions about the estimators themselves.

[0156] The belief that relationships beyond fifteen degrees were undetectable is evidenced by the fact that relationship estimators used by genetic testing companies do not even attempt to detect more distant relationships. This is because the genetic testing companies use empirically-obtained distributions for inference Since the simulations used to obtain these distributions consider a single common ancestor for each related pair, they result in zero IBD sharing for relatives beyond around 15 degrees. Consequently, it made no sense to simulate more distant relationships.

[0157] The implication of the embodiments herein is that distant relationship estimates reported by genetic testing are simply wrong. Individuals do indeed have many thousands of fifth cousins as these platforms report. However, a large proportion of relatives that are reported as fifth cousins or more are in fact much more distant. The reality is perhaps more interesting than the current incorrect estimates imply: relationships with common ancestors who lived before major world migrations such as European contact in the Americas and the Transatlantic slave trade can be detected.D. Example Operations

[0158] Figure 5 is a flow chart illustrating an example embodiment. The process illustrated by Figure 5 may be carried out by a computing device, such as computing device 100, a cluster of computing devices, such as server cluster 200, and / or a genetic computing platform. However, the process can be carried out by other types of devices or device subsystems. For example, the process could be carried out by one or more commercial cloudbased computing platforms.

[0159] The embodiments of Figure 5 may be simplified by the removal of any one or more of the features shown therein. Further, these embodiments may be combined with one another, as well as features, aspects, and / or implementations of any of the previous figures or otherwise described herein.

[0160] Block 500 of Figure 5 may involve obtaining first genetic sequence data of a first individual and second genetic sequence data of a second individual, wherein the first individual and the second individual are selected conditional on them sharing at least a threshold amount of identical -by-descent (IBD) segments of genetic sequence data.

[0161] Block 502 may involve, based on the first genetic sequence data and the second genetic sequence data, determining that the first individual and the second individual share N IBD segments of genetic sequence data, wherein the N IBD segments have respective lengths.

[0162] Block 504 may involve identifying a plurality of candidate shared ancestor sets for the first individual and the second individual, wherein each of the candidate shared ancestor sets includes a respective one or two possible shared ancestors with a respective first number of meioses between the first individual and the respective one or two possible shared ancestors and a respective second number of meioses between the second individual and the respective one or two possible shared ancestors.

[0163] Block 506 may involve for each of the candidate shared ancestor sets, determining probabilities that the N IBD segments of the respective lengths would occur given that the first individual and the second individual share at least one IBD segment.

[0164] Block 508 may involve based on the probabilities, selecting one of the candidate shared ancestor sets as most-recent IBD-contributing shared ancestors of the first individual and the second individual.

[0165] In some implementations, selecting one of the candidate shared ancestor sets as the most-recent IBD-contributing shared ancestors is also based on the respective lengths of a set of the N IBD segments.

[0166] In some implementations, selecting one of the candidate shared ancestor sets as the most-recent IBD-contributing shared ancestors is also based on a total length of the N IBD segments.

[0167] In some implementations, selecting one of the candidate shared ancestor sets as the most-recent IBD-contributing shared ancestors is also based on ranges of a total length of the N IBD segments mapped to relationship degrees.

[0168] In some implementations, the threshold amount of IBD segments is one IBD segment.

[0169] In some implementations, N is between 1 and 60, inclusive.

[0170] In some implementations, the respective first number of meioses and the respective second number of meioses are different.

[0171] Some implementations further involve, based on IBD segments shared by the first individual, the second individual, and the candidate shared ancestor that was selected, determining a likelihood of a disease for the first individual or the second individual, wherein the disease has a genetic basis in at least one of the IBD segments shared.

[0172] Some implementations further involve, based on IBD segments shared by the first individual, the second individual, and the candidate shared ancestor that was selected, determining a time of divergence or merger of a genetically isolated population.

[0173] The information determined by any step in the process of Figure 5 - including intermediate steps - may be displayed on a graphical user interface. Some examples follow.

[0174] One possible graphical user interface display for representing a shared ancestor set containing the most recent IBD-contributing shared ancestors of two individuals is an interactive pedigree chart or family tree visualization. In this graphical representation, ancestors can be arranged in generational layers, with each ancestor node labeled with identifying information (e.g., name, birth / death dates). Visual cues, such as color-coding or highlighting, can indicate ancestors who contributed genetic segments directly identifiable as IBD between two selected individuals. Users could interact with this display by clicking or hovering over ancestor nodes to expand further details, such as genetic markers, recombination breakpoints, or IBD segment lengths.

[0175] Another possible graphical user interface display is a timeline-based visualization of ancestors and corresponding IBD segments. In this approach, ancestors can be plotted chronologically along a horizontal axis representing generations, with vertical markers indicating the presence and length of IBD segments inherited from these ancestors. This display could utilize a heatmap-style gradient, where segment lengths or number of segments inherited are reflected by varying color intensities or thicknesses of graphical elements. Clicking or hovering over or selecting ancestors on the timeline would reveal detailed pop-up annotations, including segment locations, genomic coordinates, or clinical annotations associated with these segments.

[0176] Accordingly, some implementations further involve providing, for display on a graphical user interface, a tree-based representation of: (i) individuals including the first individual, the second individual, and the candidate shared ancestor that was selected, and (ii) generational relationships therebetween, wherein nodes of the tree-based representation display or can be actuated to display identifying information, genetic markers, or IBD segment data relating to the individuals.

[0177] Some implementations further involve providing, for display on a graphical user interface, a timeline-based representation of: (i) individuals including the first individual, the second individual, and the candidate shared ancestor that was selected, and (ii) presence or length of IBD segments of shared by the individuals, wherein icons display or can be actuatedto display identifying information, genetic markers, or IBD segment locations relating to the individuals.

[0178] These procedures also exhibit and / or facilitate a number of scientific, technical, and / or medical advances. The advantages discussed herein are not comprehensive and other advantages may exist.

[0179] As one example, IBD segments shared by two individuals trace back to a common ancestor who transmitted the same genetic segment to their descendants. Because recombination breaks down genetic material over generations, longer shared IBD segments typically point toward a relatively recent common ancestor. Identifying this ancestor helps researchers narrow the genomic regions potentially harboring disease-related genes, allowing them to focus on a smaller, well-defined genomic area for detailed investigation. Also, precise determination which ancestor(s) contributed the IBD segment enables the formation of more accurate linkage models between IBD segments and diseases.

[0180] As another example, identifying the most-recent IBD-contributing shared ancestors between two individuals advances population genetics by enhancing analyses of population structure, admixture, and human migration patterns. Identifying recent IBD- contributing ancestors between individuals within and across populations provides granular insights into the genetic relationships that define and differentiate populations. Because recombination events occur at predictable rates over generations, the size and distribution of IBD segments can help researchers pinpoint precisely when populations diverged or merged. Further, by identifying the most-recent shared ancestors responsible for specific IBD segments, scientists can improve mapping analyses of admixed populations (e.g., those with descendants from two or more historically isolated populations). This allows accurate estimation of the timing and extent of admixture events, pinpointing historical periods when populations mixed. Additionally, tracing ancestors who contributed identifiable genetic segments enables researchers to reconstruct historical migration events with improved clarity. Precise knowledge of shared ancestral relationships and IBD segment lengths can help estimate when, from where, and along what pathways human populations migrated, interacted, and interbred.

[0181] Figure 6 is a flow chart illustrating another example embodiment. The process illustrated by Figure 6 may be carried out by a computing device, such as computing device 100, a cluster of computing devices, such as server cluster 200, and / or a genetic computing platform. However, the process can be carried out by other types of devices or device subsystems. For example, the process could be carried out by one or more commercial cloudbased computing platforms.

[0182] The embodiments of Figure 6 may be simplified by the removal of any one or more of the features shown therein. Further, these embodiments may be combined with one another, as well as features, aspects, and / or implementations of any of the previous figures or otherwise described herein.

[0183] Block 600 of Figure 6 may involve obtaining first genetic sequence data of a first individual and second genetic sequence data of a second individual, wherein the first individual and the second individual are selected conditional on the first individual and the second individual sharing at least a threshold amount of identical-by-descent (IBD) segments of genetic sequence data.

[0184] Block 602 may involve, based on the first genetic sequence data and the second genetic sequence data, determining that the first individual and the second individual share N IBD segments of genetic sequence data, wherein the N IBD segments have respective lengths.

[0185] Block 604 may involve, based on the N IBD segments of genetic sequence data and for a plurality of ancestral generations of the first individual and the second individual, estimating numbers of distinct common ancestors of the first individual and the second individual for each of the ancestral generations.

[0186] Some implementations may involve, based on the numbers of distinct common ancestors as estimated, selecting one or more of the distinct common ancestors as most likely ancestors that contributed observable IBD segments to the first individual and the second individual.

[0187] In some implementations, estimating the numbers of distinct common ancestors of the first individual and the second individual for each of the ancestral generations is based on a recurrence relationship involving: an expected number of common ancestors between the first individual and the second individual in a given generation, and a number of distinct ancestors of the first individual in the given generation that are not ancestors of the second individual.

[0188] In some implementations, the threshold amount of IBD segments is one IBD segment.

[0189] In some implementations, N is between 1 and 60, inclusive.

[0190] Some implementations further involve, based on the N IBD segments shared by the first individual and the second individual, determining a likelihood of a disease for the first individual or the second individual, wherein the disease has a genetic basis in at least one of the N IBD segments.

[0191] Some implementations further involve, based on the N IBD segments shared by the first individual and the second individual, determining a time of divergence or merger of a genetically isolated population.

[0192] The information determined by any step in the process of Figure 6 - including intermediate steps - may be displayed on a graphical user interface. Some examples follow.

[0193] One possible graphical user interface display for visualizing estimated numbers of distinct common ancestors shared by two individuals across multiple ancestral generations is a bar chart interface. In this graphical format, each bar along the horizontal axis represents a single ancestral generation, progressing chronologically into the past, while the vertical axis indicates the estimated count of common ancestors in each generation. Bars can be interactively highlighted to reveal additional details, such as confidence intervals, numerical estimates, or relevant genetic statistics. This visualization allows users to identify trends in ancestral overlap, as well as periods of increased relatedness or demographic events such as bottlenecks or admixture episodes.

[0194] Another possible graphical user interface display is an interactive heatmap or matrix-style visualization. In this approach, rows could represent ancestral generations going back in time, while columns could categorize ancestors based on relatedness metrics or genetic contribution proportions. Each cell of the heatmap would visually encode the estimated number of distinct common ancestors using color intensity gradients, with deeper shades indicating greater ancestral overlap. Users could click on or hover over individual cells or rows to see detailed annotations, such as the generation-specific count, genetic segment characteristics, or demographic interpretations.

[0195] Accordingly, some implementations further involve providing, for display on a graphical user interface, a bar chart, wherein each bar along a horizontal axis represents a single ancestral generation, progressing chronologically into the past, and wherein a vertical axis represents an estimated count of the distinct common ancestors in each of the ancestral generations.

[0196] Some implementations further involve providing, for display on a graphical user interface, a heatmap, wherein rows of the heatmap represent generations progressing chronologically into the past, and wherein columns categorize the distinct common ancestors based on relatedness metrics or genetic contribution proportions.

[0197] These procedures exhibit and / or facilitate a number of scientific, technical, and / or medical advances. The advantages discussed herein are not comprehensive and other advantages may exist.

[0198] As one example, by quantifying how many distinct ancestors two individuals share at each generation, scientists can more accurately model their degree of genetic relatedness. This detailed understanding directly informs predictions about shared genetic risks. For instance, accurately measuring relatedness can pinpoint genetic risk factors for inherited diseases, especially those following recessive inheritance patterns. By knowing the extent and distribution of ancestral overlap, clinicians can better estimate disease risks and tailor genetic counseling accordingly.

[0199] As another example, knowledge about the number of distinct shared ancestors per generation can refine demographic models and historical population reconstructions. This can enable researchers to infer more precise timelines of demographic events such as population expansions, migrations, or bottlenecks. By understanding when the number of shared ancestors sharply increases or decreases, researchers can map historical population shifts.

[0200] In still another example, the frequency and distribution of shared ancestors can be used to determine how certain genetic segments are maintained or eliminated by natural selection. Insights about shared ancestral frequency across generations help researchers detect signatures of natural selection or genetic drift by observing how segments inherited from common ancestors persist or disappear. For instance, when certain IBD segments derived from particular ancestors persist unusually long across multiple generations in descendants, this could indicate selective pressure preserving beneficial genetic variants.IV. Improved Simulation of Pedigrees

[0201] An algorithm for simulating meioses within a pedigree structure is presented. The pedigree structure is conditional on the event that certain individuals share IBD with one another, and has utility in two common simulation problems.

[0202] Sets of individuals ascertained on the basis of IBD sharing can be used in downstream analyses that require the simulation of recombination events within the pedigree relating the individuals. Such simulations can be used to generate training or testing data for statistical methods Simulations may also be performed in the case of family or linkage analyses to assess power (e.g., how many individuals should be considered to confidently detect a shared IBD segment). Simulations also underlie the likelihoods used by many relationship inference methods. In all of these cases, failure to simulate relationships and pedigrees under the proper model of ascertainment can lead to bias.

[0203] Many methods have been developed for simulating meioses (in sexually reproducing organisms, the production of gametes with half as many chromosomes as thestarting cell) within a pedigree structure. However, no existing method makes it possible to condition on the event that two or more individuals share IBD.

[0204] In cases in which relatives are closely related, one can use an existing simulation method together with rejection sampling to obtain samples under the conditional distribution by rejecting any sample in which the required individuals share no IBD. However, when relatives are distantly related, or when multiple pairs of individuals are moderately-to-distantly related, rejection sampling becomes impractical because the fraction of rejected samples is close to one. Therefore, an alternative method for conditional sampling is desirable.

[0205] One clear use case for an efficient conditional sampler is the simulation of IBD- sharing distributions for relatives of increasing degree. These distributions are used in relationship inference methods such as those employed by genetic testing companies. At present, the distributions used by these inference methods rely on unconditional distributions; however, the conditional and unconditional distributions diverge substantially for relationships beyond fourth cousins, resulting in dramatically biased relationship estimates.

[0206] A second use case is the simulation of individuals and their set of moderate-to- distant relatives. In large genotyping datasets such as the UK Biobank or other datasets, individuals have many thousands of relatives. Moreover, each individual shares many pairs of common ancestors with each of these relatives, making it likely for distant relatives to share IBD through some common ancestor. As a result, each individual shares IBD with many thousands of other individuals at various degrees of relatedness.

[0207] These large sets of relatives jointly carry information that is useful for inference. Tools that leverage these sets of relatives to perform inference include linkage methods, familybased association methods, pedigree inference methods, local ancestry inference methods, clustering methods, and methods that infer genotypes in un-genotyped ancestors, among others. All of these use cases benefit from the ability to simulate the large pedigrees relating these individuals for purposes such as inference method testing, power analyses and the empirical sampling of distributions that are difficult to obtain analytically. Rejection sampling cannot be used to simulate IBD between moderate to distant relatives because the probability of rejection is nearly one.

[0208] The embodiments herein show how to sample meioses and transmitted IBD among pairs of individuals and within a pedigree, conditional on the event that two or more individuals share IBD with one another. The utility of this simulator is demonstrated for two downstream applications: pedigree inference and the simulation of an individual together with their set of distant relatives.A. Sampling Conditional on Any Observed IBD

[0209] Suppose two individuals ( i and j ) are related through relationship R and suppose it is known that they share some total, but unspecified, amount of IBD I . The amount they share may simply be nonzero (J > 0) or it may be greater than some predetermined threshold T. A goal is to simulate recombination events on the ancestral path connecting i and j, conditional on the event that / > T > 0. This can be scaled up to sample meioses in a pedigree in which arbitrary sets of individuals are known to share IBD with one another.

[0210] Put another way, a goal is to specify a relationship R — (u, d, a) between i and j, where it is the number of meioses “up” from i to their common ancestor(s) with j and d is the number of meioses “down” from the common ancestor(s) to j. The number of common ancestors is a, which is either 1 or 2 in an outbred pedigree. Although inbreeding is common in many populations and although all human pedigrees contain loops when a sufficient number of generations in the past are considered, the focus here is on outbred pedigrees here because it makes the analysis simpler. The model can be extended to include consanguineous relationships by including considerably more bookkeeping. The method is also slower in the case of inbreeding because of the need to sample larger groups of meioses at the same time to satisfy constraints.

[0211] Figure 7 depicts a pedigree of individuals (circles) with meioses (lines) between them. Nodes i and j are related through common ancestors x and y. The number of meioses “up” from i to its common ancestors with j is u = 3. The number of meioses down from the common ancestors to j is d = 2. The number of common ancestors is a = |{x, y}| = 2. This yields the relationship R = (3, 2, 2) between i and j.

[0212] As noted, a goal is to sample all it meioses up from i to its first common ancestor with j and all d meioses down from its first common ancestor to j (e.g., darker meioses in Figure 7). If i and j have two common ancestors (a = 2) then two additional meioses (e.g., lighter meioses in Figure 7) should be sampled, leading to their second shared common ancestor.

[0213] To sample conditionally on the event that i and j share at least t centimorgans of DNA IBD with one another, Gibbs sampling can be used in which each meiosis is sampled sequentially in turn, conditional on all the other meioses. The case of transmission for a single chromosome is first considered and then it is generalized to the whole genome.

[0214] Gibbs sampling may include a Markov chain Monte Carlo algorithm used to approximate complex multivariate probability distributions by iteratively sampling fromconditional distributions. It is especially useful when direct sampling from the joint distribution is difficult. In some cases, Gibbs sampling may not be necessary - only a forward pass and backward pass along each path between two genotyped may be required. However, formulating this technique as a Gibbs sampler makes it clear that the sampler obtains the correct distribution.

[0215] For meiosis m, let bmbe the sorted list of recombination breakpoints (specific locations on chromosomes where breaks and subsequent recombinations occur during the process of meiosis) among the two parental haplotypes undergoing recombination for a particular chromosome and let hmE {0, 1} be the parental haplotype on which copying starts. In a diagram of the pedigree, the convention is that hm= 0 corresponds to the haplotype of the “left” grandparent and hm= 1 corresponds to the haplotype of the “right” grandparent. For example, in the meiosis between child c and parent p in Figure 7, haplotype 0 in p is the one inherited from x, whereas haplotype 1 in p is the haplotype inherited from y. In general, the lexicographically first parent label corresponds to the “left” parent.1. Sampling for any Transmitted IBD

[0216] Gibbs sampling can be used to sample the list( / i2, b2) ... hM, bM)] of at least some of the M = u + d + 2(a — 1) meioses between a pair of individuals by sampling each tuple (hm, bm) sequentially, one at a time, conditional on the breakpoints sampled at all other meioses.

[0217] For any given meiosis m, an individual meiosis (hm, bin) is sampled using a rejection sampling approach or another technique. For example, using rejection sampling, hmis sampled from a Bernoulli random variable; then, the number nm— dim(bm~) of breakpoints are sampled from a Poisson random variable with mean L / 100, and the positions of each of the nmbreakpoints is sampled uniformly on the interval [0, L], where L is the length of the chromosome in centimorgans.

[0218] Starting with unconditionally sampled breakpoints and starting haplotypes at each meiosis, samples are rejected until a sample is obtained in which segments transmitted from i overlap by at least T centimorgans with segments transmitted from j. Once this result is achieved, the proposed tuple (hm, bin) is accepted and the process moves on to sample

[0219] Acceptance rules when a segment can be transmitted. To illustrate the acceptance rule, a single meiosis in shown in panel A of Figure 8. In this meiosis, a child haplotype c inherits from the two parental haplotypes p0and p, of parent p. A focus is onsampling the starting haplotype h and the breakpoints b conditional on the event that some IBD is transmitted between relatives rl and r2.

[0220] In panel A of Figure 8, arrows show the locations of breakpoints (b) and the ancestral haplotype that is copied from to the right of each breakpoint. Dotted lines show key locations along the chromosome for easier visualization. From left to right, copying starts from ancestral haplotype p0(event h — 0), then switches to ancestral haplotype p}and then back to haplotype p0. The highlighted region on c shows genetic material from i, a descendant of c. Highlighted regions on p0and p±show genetic material from j, a relative of p.

[0221] A meiosis (h,b) successfully transmits IBD between i and j if the highlighted segments in c overlap with the highlighted segments in p on a haplotype that is transmitted from p to c. These transmitted haplotypes are shown in the bottom row in panel B of Figure 8. When the total length of the transmitted haplotype segments is greater than T, the proposed tuple (hm, bin) is accepted.

[0222] Aversion of this sampling approach can also be used to sample from a genetic map. The genetic map approach can be used to sample from the full genome. Both of these possibilities are discussed below.

[0223] Acceptance rules when multiple meioses are sampled together In some cases, IBD between two individuals can be propagated along two different paths in the pedigree. A common example of this case is when two individuals share IBD through a pair of common ancestors. This scenario shown in Figure 7, in which i and j can share IBD either along the path through common ancestor x or common ancestor y, or through both.

[0224] In the case of Figure 7, an acceptance criterion that requires IBD to be transmitted between i and j separately through each of the meioses (p, x) and (p, y) cannot be used. Doing so would require IBD to be transmitted along both paths, which is more restrictive than the criterion that IBD is transmitted at all.

[0225] Instead, both meioses (p, x) and (p,y) can be jointly sampled and a proposed sample can be accepted as a success if either (p,x) or (p, y) is a success. Moreover, it can dramatically increase the rate of acceptance if all meioses (c,p), (p, x), (p,y), (z, x), (z,y), and 0',z) are jointly sampled, which ensures that IBD from i and j is passed to the proper parental haplotypes in x and y. If c and j are not included in the sampling block, then meioses could be accepted in which all segments from i are shared on the haplotype in p that comes from y and all segments from j are shared on the haplotype of z that comes from x. Such a case would not permit sampling meioses (p, x), (p, y), (z, x), and (z,y) in which segmentsfrom i are shared with j through p and z.

[0226] In general, sampling is most efficient if all meiosis pairs that form a block in which constraints must be simultaneously satisfied are jointly sampled. Such blocks can be assembled by the following rules: (1) if meiosis (c, p) is in the block then all meioses involving full siblings of c must be in the block, and (2) if meiosis (c, p) is in the block then all meioses between p and children of p must be in the block. More general forms of block sampling can be used to handle more complicated loops. In such blocks, meioses can be sampled along all divergent paths in the loop In principle, this would allow sampling from inbred pedigrees. The longer the path, the more unlikely it is that IBD will be transmitted through it, making the simultaneous sampling of multiple paths increasingly less practical as the paths become longer.

[0227] Acceptance rules when no segment can be transmitted. Note that in sampling meiosis (c, p) between child c and parent p in Figure 7, the other meoises in the pedigree may have been sampled in such a way that no segments from i have been propagated between i and c and / or no segments from j have been propagated between p and j.

[0228] When it is not possible to transmit r centimorgans of IBD, a tuple ( / im, bm) can still be accepted. Acceptance can be performed in such a way that it helps propagate segments through the pedigree even when it is impossible to fully transmit a segment between two genotyped individuals. This leads to generalized acceptance rules: a proposed sample (h.m, bm) is accepted if (1) No segments are available either above or below to transmit; so transmission is impossible; (2) At least T centimorgans of segments are available from below to transmit and at least T centimorgans of these segments are transmitted up; (3) at least T centimorgans of segments are available from above to transmit and at least T centimorgans of these segments are transmitted down; (4) at least T centimorgans of segments are available both above and below to transmit and at least T centimorgans of these segments are transmitted both down and up.

[0229] This rule for accepting proposed samples when it is not possible to fully transmit at least T centimorgans of IBD has the purpose of speeding up the transmission of IBD through the pedigree until all required IBD has been transmitted. Once IBD has been transmitted among all IBD-sharing pairs, proposed samples ( / ican be accepted under the correct condition that at least T centimorgans of IBD is propagated.

[0230] In general, when sampling sets of meioses together, rather than one at a time, the above criteria are applied to the “receiving’ and “transmitting” nodes in the group, rather than the child and parent nodes. For instance, if a group consists of nodes p, x, y and z inFigure 7, then node z is the node in the group that “transmits” segments from j and node p is the node that “receives” these segments and passes them on to i. Thus, the following should be true in order to accept a sample jointly from (p,xft (p,y), (z.xft and (z, y): (1) if nodes i and p share at least T centimorgans, then at least t centimorgans must be transmitted to z (for propagation efficiency); (2) if nodes j and z share at least T centimorgans, then at least t centimorgans must be transmitted to p (for propagation efficiency); (3) if nodes i and p share at least T centimorgans and if j and z share at least T centimorgans and if at least T centimorgans overlaps between these segments on a meiosis path, then at least T centimorgans of this overlap must be transmitted. Note that this criterion requires the overlap to occur on a feasible meiosis path. If a segment in z physically overlaps a segment in p, but on different parental haplotypes, then the transmission is impossible and the proposed sample can be accepted.2. Data Structures

[0231] In order to establish constraints on each meiosis, a number of data structures may be employed. The first data structure, ibd dict, records the segments propagated at each sampling step between each genotyped node in the pedigree and each haplotype of each other node in the pedigree. This dictionary is updated at each sampling step. Another data structure, constraint diet records the pairs of genotyped IDs that must transmit IBD through each haplotype of each other node in the pedigree, along with the direction of the transmission; for example, for the pedigree in Figure 7 constraint dict records that IBD is passed from node i to node j “up” through haplotype 0 of node c. This dictionary can be established before sampling begins and does not change. Another two dictionaries, start_hap_dict and bpt_pos_list_dict, store the starting haplotypes and breakpoint positions in each meiosis. These dictionaries can be updated on each sampling step. Finally, the pedigree structure itself is stored in a dictionary, up dict of the form {c : {pi.Pz}, ■ ■ ■ } mapping each child node c to zero, one or two parent nodes. It is also convenient to store the reverse (down_dict) of up_dict mapping parents to their child nodes.3. Sampling with a Genetic Map

[0232] denote a genetic map from physical positions (denoted / j) togenetic positions gt. The valuesare strictly increasing in i and the map from f to g is monotonically non-decreasing. Note that the genetic distance from Tosample from such a map, the number of breakpoints in the region (Jftft+i) can be sampled from a Poisson distribution with mean e positions of these breakpoints inthe region are then uniformly distributed over the interval ( / ), / i+i). Each region is sequentially sampled from left to right along the chromosome. The starting haplotype is sampled according to a Bernoulli distribution.4. Sampling a full genome with any transmitted IBD

[0233] Sampling a full genome is more complicated than sampling each chromosome independently because some IBD is guaranteed to be transmitted using the technique herein. Thus, applying it to each chromosome separately would yield IBD on each chromosome. A simple work-around to this problem is to sample from the whole genome all at once by modeling it as a single, long chromosome amounting to all chromosomes concatenated end-to- end with gaps between them. Over this long chromosome, a mask is applied to ignore samples within the regions between chromosomes. The genetic map is set up so that the gaps between chromosomes correspond to long genetic distances, thereby allowing essentially free recombination in these regions.

[0234] To that end, Figure 9 depicts sampling of two chromosomes, each of physical length lOOMB and genetic length 100 centimorgans, by simulating one long chromosome comprised of the two chromosomes concatenated together with a gap between them (the region in the rectangle). All segments and transmissions in the gap are ignored and only transmissions that occur outside of it are accepted. Regardless of the physical length of the gap, the genetic length of the gap can be set to a large number to approximate free recombination between the two chromosomes. In Figure 9, the gap is 1MB long in physical coordinates, but 1,000 centimorgans long in genetic coordinates. The approach can be extended to arbitrarily many chromosomes with the caveat that sampling in the masked regions requires compute time.

[0235] Another more efficient approach is to sample all chromosomes at the same time in a meiosis (i.e., without concatenating and masking) and accept a set of proposed breakpoints if at least T centimorgans of IBD is transmitted in the full genome. This approach has the advantage that sampling segments within masked regions is not needed. Ultimately, either approach is efficient since they ignore masked segments. Masking is useful for ignoring chromosomal regions, such as those with high levels of background IBD.5. Comparison with Rejection Sampling

[0236] To check that the conditional sampler is producing the correct distribution, it was compared with simple rejection sampling for scenarios in which rejection sampling was computationally tractable.

[0237] Figure 10 depicts a comparison of the simple rejection sampler with the Gibbs sampler for different relationships R = (u, d, a). 1,000 replicates of each relationship weresampled using each method. Both samplers conditioned on the event that T > 0 centimorgans of IBD was transmitted between the two relatives. The simulations are based on one chromosome of length 100 centimorgans. Each panel shows a histogram based on 1,000 samples. For each sample, the Gibbs sampler made 10 passes, sampling each group of meioses ten times. The event on which this was conditioned was T > 0 centimorgans: i.e., that any IBD was transmitted between the two relatives.

[0238] Figure 10 shows that the conditional Gibbs sampler produces the same segment count and length distributions as the simple rejection sampler, but that the relative number of Gibbs replicates becomes increasingly smaller relative to the number of simple rejection replicates as the degree of the relationship becomes larger.

[0239] Figure 11 depicts the timing of the simple rejection sampler compared with timing of the Gibbs sampler. Simulations were performed on a pair of nth full cousins whose common ancestors lived g generations in the past. Each datapoint is the mean time over 100 replicates and error bars show one standard deviation above and below the mean. Simulations were performed only for g < 13 for the simple rejection sampler, after which point rejection sampling became difficult.

[0240] From Figure 11, it can be seen that the overall timing for the Gibbs sampler stays manageable even beyond 30 generations in the past, at which point the probability of observing IBD between two genealogical relatives becomes low. Although the number of required replicates is higher for the Gibbs sampler when the degree of the relationship is small, it quickly becomes small relative to the number of replicates required for full rejection sampling. The relative computation time of the two samplers is also shown in Figure 11.

[0241] Sampling on a pedigree. The Gibbs conditional sampling approach becomes especially useful when there is more than one pair of relatives who must share IBD with one another. In this case, the probability that all constrained pairs share IBD at once can become exceedingly small, making simple rejection sampling impractical even for small pedigrees.

[0242] Figure 12 depicts a comparison of the rejection sampler and the Gibbs sampler for a pedigree with multiple constrained pairs. Panel A of Figure 12 depicts the pedigree structure. Individuals i, j, and k are related such that i and j share two common ancestors and k shares a single common ancestor with each of i and j. Panel B of Figure 12 depicts timing for the rejection and Gibbs samplers for different values of u, u', and d. In each pedigree, u' was constrained to 2, it = d = g, and g was varied from 1 to 10.

[0243] In contrast with the analysis shown in Figure 10, conditioning is now based onthe much more restrictive event that three different pairs of genotyped individuals share IBD with one another. Specifically, the event that i shares IBD with j, j shares IBD with k, and i shares IBD with k.

[0244] Panel C of Figure 12 shows a comparison of the distributions of segment counts and lengths for individuals i and j. This is easier to visualize than the full joint distribution of segments shared between all pairs of genotyped nodes i, j, and k. Two statistics, the number of segments and segment lengths, are shown between i and j for relationship types R = (2, 2, 2) (first cousins) and R = (5, 5, 2) (fourth cousins). By comparing the segment count distribution for the relationship R = (2, 2, 2) in Panel C of Figure 12 with the count distribution for this relationship in Figure 10, it can be seen that these are not the same distributions due to the additional constraint that i and j share IBD with k as well as with one another.

[0245] By comparing Panel B of Figure 12 with Figure 11, it can be seen that including the additional requirement that IBD is shared with a third individual increases the computational time of both the simple rejection sampler and the Gibbs conditional sampler, but that it affects the simple rej ection sampler much more than the conditional sampler. In particular, the Gibbs sampler becomes computationally more efficient than the simple rejection sampler when the common ancestors are around g = 4 generations in the past, compared with g = 7 generations in the past for the two-relative case shown in Figure 11. In the more highly constrained case of a three-person pedigree, the simple rejection sampler becomes unwieldy by 9 = 7 generations in the past, compared to g = 13 generations in the two-person case. In contrast to the rejection sampler, the Gibbs sampler can be used for quite distant relationships.

[0246] Although one can run rejection simulations in parallel, the expected amount of shared IBD decreases exponentially with each degree of separation between two IBD-sharing individuals or with the addition of IBD-sharing pairs. As a result, one cannot simply run jobs in parallel to overcome the computational slowness of rejection sampling. For instance, the probability that two individuals share any IBD when they are separated through a single common ancestor who lived 15 generations in the past is approximately 5 x 10-7and when the ancestor existed 20 generations in the past this probability is approximately 6 X 10-10.

[0247] Adding in additional constrained pairs quickly makes the problem completely intractable. For instance, the lineage extending upward from 1 to 16 via nodes -2 and -5 in Figure 14 is nearly independent of the lineage extending upward from nodes 1 to 10 through nodes -2 and -4. To jointly sample IBD sharing among nodes 1 and 16 and 1 and 10, onewould to satisfy these two nearly independent constraints, each of which has a probability of < IO-3. From this, it is clear that the sampling problem quickly becomes computationally intractable after the addition of a few IBD-sharing pairs.B. Applications1. Simulating IBD-Sharing Distributions for Relationship Inference

[0248] The conditional sampler described above can be used for simulating IBD between distant relatives who are known to share some amount of IBD. An immediate application of this sampler is simulating IBD sharing distributions that can be used in relationship estimators. As noted, the unconditional distributions that are currently used for this purpose are inappropriate because they do not account for the fact that putative relatives are generally ascertained on the basis of IBD sharing with one another.

[0249] For relationships of increasingly distant degree, Panel A of Figure 13 shows the mean total IBD between two relatives separated by various degrees, including the mean IBD from the unconditional distribution and the mean IBD from the conditional distribution is shown in orange. From Panel A of Figure 13, it can be seen that the two means begin to diverge when relationships are approximately 8 degrees (approximately third cousins). For relationships ten degrees and greater (approximately fourth cousins) the distributions have largely diverged. The unconditional mean total IBD goes to zero quickly and cannot be shown on the log-scale plot in for degrees above approximately 20.

[0250] The discrepancy between the conditional and unconditional distributions has major implications for relationship inference. Panel B of Figure 13 shows degree estimates from a very simple method of moments estimator. The estimator takes the total IBD observed for a pair of individuals and finds the degree for which the expected total length of IBD most closely matches the observed total IBD. This kind of estimator is similar to those used by several direct-to-consumer genetic testing companies and bias engendered by employing the unconditional distribution is found in all currently-used relationship estimators.

[0251] Thus, panel B of Figure 13 shows that the estimator based on the unconditional distribution is heavily biased and reaches an asymptote near ten degrees. This explains the fact that the most distant relationships inferred by existing estimators are around ten degrees. In contrast, the unconditional estimator makes it possible to infer much greater degrees, although the moment estimator is still biased. The implication of Figure 13 is that one must use conditional sampling to obtain empirical distributions that are used for relationship inference in the usual case in which putative relatives are ascertained on the basis of IBD sharing.2. Simulating the Pedigrees of Today

[0252] In datasets that contain many genotyped or sequenced individuals, like the UK Biobank and direct-to-consumer genetic testing databases, genealogical relationships can be detected that extend tens of generations into the past. There is a need to be able to simulate data for these kinds of pedigrees in order to support the development of new algorithms. The conditional sampler presented herein makes it possible to simulate IBD between an individual and a set of distant relatives. Figure 14 shows one such pedigree, including an individual and twenty of their distant relatives, each separated from the focal individual by 20 degrees of relatedness. Each individual is denoted by a disc and each line represents a connection between two individuals Genotyped nodes are indicated with positive numbers and ungenotyped nodes are indicated with negative numbers. In this topology, a focal individual (1) is connected to their distant relatives, each through a single common ancestor who lived ten generations in the past.

[0253] This pedigree was simulated in under a minute using the conditional sampler. A single lineage of this pedigree would require an average of > 103replicates to achieve an acceptance from the rejection sampler and all 20 lineages together would require ~ IO60replicates, rendering the simulation effectively impossible using existing simulation methods.

[0254] In other words, existing techniques for simulating such a pedigree would require so much processing and / or memory capacity that they are practically impossible to perform (e.g., taking days, weeks, months, years, or even billions of years of computing time). The embodiments herein, however, can provide such a simulated pedigree based on conditional sampling within seconds or minutes. Therefore, these embodiments result in a technical improvement to a technical problem by dramatically reducing the need for computer resources.

[0255] Methods like the one presented here are usable to simulate the kinds of pedigrees that appear in modem genotyping datasets. As these databases increase in size, simulation methods will need to handle increasingly large and dense pedigrees. Several aspects of the existing simulator can be improved in order to decrease the runtime. In particular, the rejection component of the sampler that samples within a single meiosis does not take into account the regions in which samples must occur and therefore, while providing a significant improvement over prior techniques, is non-optimal. For any sampled starting haplotype, there are further constraints on the number and location of breakpoints that can substantially speed up sampling.

[0256] Another speed-up that reduces computing resource usage can come from rearranging the order in which meioses (and blocks of meioses) are sampled. The order used in the sampler presented here is arbitrary; however, if sampling proceeds recursively throughthe pedigree, sampling each meiosis or meiosis-block only after the neighboring block has been sampled, then IBD can be spread efficiently through the pedigree and the data structure ibd dict can be updated efficiently by updating entries using entries from neighboring nodes.

[0257] Using IBD sharing statistics simulated with this approach, it is possible to replace the biased distributions that are still used for relationship inference with distributions that are appropriate for the inference problem. Doing so has a considerable effect on the distant relationships inferred in large genotyping databases, moving them from five or six generations in the past to fifteen or twenty generations in the past. The variance in these estimates also increases substantially, reflecting the true underlying uncertainty.C. Example Operations

[0258] Figure 15 is a flow chart illustrating an example embodiment. The process illustrated by Figure 15 may be carried out by a computing device, such as computing device 100, a cluster of computing devices, such as server cluster 200, and / or a genetic computing platform. However, the process can be carried out by other types of devices or device subsystems. For example, the process could be carried out by one or more commercial cloudbased computing platforms.

[0259] The embodiments of Figure 15 may be simplified by the removal of any one or more of the features shown therein. Further, these embodiments may be combined with one another, as well as features, aspects, and / or implementations of any of the previous figures or otherwise described herein.

[0260] Block 1500 of Figure 15 may involve determining a relationship between a first individual and a second individual, wherein the relationship includes one or two possible shared ancestors of the first individual and the second individual, wherein a first number of meioses is between the first individual and the one or two possible shared ancestors, and wherein a second number of meioses is between the second individual and the one or two possible shared ancestors.

[0261] Block 1502 may involve determining that the first individual and the second individual share at least one IBD segment of genetic sequence data.

[0262] Block 1504 may involve probabilistically sampling a set of recombination breakpoints of each of the meioses sequentially, conditioned on any previously-sampled recombination breakpoints and conditioned on at least one IBD segment being shared between the first individual and the second individual, wherein the set of recombination breakpoints is accepted when it meets an acceptance criterion.

[0263] In some implementations, the one or two possible shared ancestors is one shared ancestor.

[0264] In some implementations, the one or two possible shared ancestors are two shared ancestors.

[0265] In some implementations, the first number of meioses and the second number of meioses are different.

[0266] In some implementations, the recombination breakpoints comprise locations on chromosomes where breaks and subsequent recombinations occur during the meioses.

[0267] In some implementations, probabilistically sampling the set of recombination breakpoints comprises Bernoulli sampling of a parental haplotype on which copying for a meiosis starts.

[0268] In some implementations, probabilistically sampling the set of recombination breakpoints comprises Poisson sampling of a number of the recombination breakpoints of a meiosis.

[0269] In some implementations, probabilistically sampling the set of recombination breakpoints comprises the recombination breakpoints of a meiosis being sampled uniformly along a chromosome.

[0270] In some implementations, the acceptance criterion comprises the set of recombination breakpoints including IBD segments of a meiosis overlapping by a threshold length for the first individual and the second individual.

[0271] Some implementations may further involve, based on the set of recombination breakpoints, deriving IBD sharing distributions for relationship estimation conditioned on shared IBD segments.

[0272] Some implementations may further involve, based on the set of recombination breakpoints, deriving a pedigree for a plurality of individuals and relationships therebetween conditioned on shared IBD segments.

[0273] In some implementations, the plurality of individuals includes 20 or more individuals, and wherein at least some of the plurality of individuals are separated by 10 or more degrees of relatedness.

[0274] The information determined by any step in the process of Figure 15 - including intermediate steps - may be displayed on a graphical user interface. Some examples follow.

[0275] One possible graphical user interface could display an interactive family tree diagram that visually represents the relationship between two or more individuals, with nodes representing each individual, connecting lines indicating the meioses from these individuals toone or two possible shared ancestors, and color-coded markers to denote the presence of shared IBD segments. Clicking on or hovering over nodes or lines can reveal detailed information such as the number of meioses and specific recombination breakpoints.

[0276] Another possible graphical user interface could involve a split-panel display where one panel provides a dynamic pedigree chart mapping out familial connections between two or more individuals and their shared IBD segments, while an adjacent panel offers a sequential, timeline-style visualization that charts recombination events across meioses. In this arrangement, users could interact with the timeline to view the recombination breakpoints, allowing for user-driven adjustments based on meiosis counts and probability thresholds.

[0277] Accordingly, some implementations may further involve providing, for display on a graphical user interface, an interactive family tree that visually represents relationships between individuals including the relationship between the first individual and the second individual, wherein nodes in the interactive family tree represent each individual, wherein connecting lines indicate meioses from the first individual and the second individual to the one or two possible shared ancestors, and wherein coded markers represent presence of shared IBD segments.

[0278] Some implementations may further involve providing, for display on a graphical user interface, a split-panel display, wherein one panel provides a pedigree chart representing familial connections between individuals including the relationship between the first individual and the second individual and their shared IBD segments, and wherein an adjacent panel provides a sequential, timeline-style visualization that represents recombination events across meioses.

[0279] Figure 16 is a flow chart illustrating an example embodiment. The process illustrated by Figure 16 may be carried out by a computing device, such as computing device 100, a cluster of computing devices, such as server cluster 200, and / or a genetic computing platform. However, the process can be carried out by other types of devices or device subsystems. For example, the process could be carried out by one or more commercial cloudbased computing platforms.

[0280] The embodiments of Figure 16 may be simplified by the removal of any one or more of the features shown therein. Further, these embodiments may be combined with one another, as well as features, aspects, and / or implementations of any of the previous figures or otherwise described herein.

[0281] Block 1600 of Figure 16 may involve determining a pedigree including a set of relationships among a set of individuals, wherein each of the relationships has a respective oneor two possible shared ancestors with a respective first number of meioses between a respective first individual and the respective one or two possible shared ancestors, and wherein each of the relationships also has a respective second number of meioses between a respective second individual and the respective one or two possible shared ancestors.

[0282] Block 1602 may involve determining which of the relationships are of individuals that share at least one IBD segment of genetic sequence data.

[0283] Block 1606 may involve, for each of the relationships, probabilistically sampling a set of recombination breakpoints of each of the meioses sequentially, conditioned on any previously-sampled recombination breakpoints and conditioned on at least one IBD segment being shared between individuals of the relationship, wherein the set of recombination breakpoints is accepted when it meets an acceptance criterion.

[0284] In some implementations, the respective one or two possible shared ancestors is one shared ancestor.

[0285] In some implementations, the respective one or two possible shared ancestors are two shared ancestors.

[0286] In some implementations, the respective first number of meioses and the respective second number of meioses are different.

[0287] In some implementations, the recombination breakpoints comprise locations on chromosomes where breaks and subsequent recombinations occur during the meioses.

[0288] In some implementations, probabilistically sampling the set of recombination breakpoints comprises Bernoulli sampling of a parental haplotype on which copying for a meiosis starts.

[0289] In some implementations, probabilistically sampling the set of recombination breakpoints comprises Poisson sampling of a number of the recombination breakpoints of a meiosis.

[0290] In some implementations, probabilistically sampling the set of recombination breakpoints comprises the recombination breakpoints of a meiosis being sampled uniformly along a chromosome.

[0291] In some implementations, the acceptance criterion comprises the set of recombination breakpoints including IBD segments of a meiosis overlapping by a threshold length for the respective first individual and the respective second individual.

[0292] Some implementations may further involve, based on the set of recombination breakpoints, deriving IBD sharing distributions for relationship estimation conditioned on shared IBD segments.

[0293] Some implementations may further involve, based on the set of recombination breakpoints, deriving a pedigree for at least some of the individuals and relationships therebetween conditioned on shared IBD segments.

[0294] In some implementations, the pedigree includes 20 or more of the individuals, and wherein at least some of the individuals in the pedigree are separated by 10 or more degrees of relatedness.

[0295] The information determined by any step in the process of Figure 15 - including intermediate steps - may be displayed on a graphical user interface. Some examples follow.

[0296] One possible graphical user interface could display an interactive family tree diagram that visually represents the relationship between two or more individuals, with nodes representing each individual, connecting lines indicating the meioses from these individuals to one or two possible shared ancestors, and color-coded markers to denote the presence of shared IBD segments. Hovering over nodes or lines can reveal detailed information such as the number of meioses and specific recombination breakpoints.

[0297] Another possible graphical user interface could involve a split-panel display where one panel provides a dynamic pedigree chart mapping out familial connections between two or more individuals and their shared IBD segments, while an adjacent panel offers a sequential, timeline-style visualization that charts recombination events across meioses. In this arrangement, users could interact with the timeline to view the recombination breakpoints, allowing for user-driven adjustments based on meiosis counts and probability thresholds.

[0298] Some implementations may further involve providing, for display on a graphical user interface, an interactive family tree that visually represents relationships between the individuals, wherein nodes in the interactive family tree represent each of the individuals, wherein connecting lines indicate meioses from a pair of the individuals to the one or two possible shared ancestors, and wherein coded markers represent presence of shared IBD segments.

[0299] Some implementations may further involve providing, for display on a graphical user interface, a split-panel display, wherein one panel provides a pedigree chart representing familial connections between the individuals and their shared IBD segments, and wherein an adjacent panel provides a sequential, timeline-style visualization that represents recombination events across meioses.

[0300] Further aspects of the embodiments of either Figures 15 or 16 may involve using the second and / or third example embodiment to generate training data for a machine learning algorithm, to generate testing data for a machine learning algorithm, to generate data from adistribution for the purpose of performing a power analysis to determine the power of a statistical test or analysis, or to empirically obtain a probability distribution on the number, lengths, or physical arrangement of IBD segments

[0301] The aspects represented in Figure 15 and 16 represent a technical improvement. The embodiments herein involve a conditional sampling method that dramatically improves simulation of pedigrees, such as those encountered in large-scale genetic datasets. Unlike conventional rejection sampling techniques — which may require and impractical extent of computational resources, the embodiments herein enable rapid simulation of complex pedigree structures by efficiently sampling recombination breakpoints conditioned on previously sampled events and the presence of shared IBD segments. For example, the conditional sampler can generate a simulated pedigree comprising a focal individual and twenty distant relatives (each separated by 20 degrees of relatedness, via a common ancestor from ten generations prior) in under a minute. This substantial reduction in required processing power and runtime not only overcomes existing limitations but also provides a scalable solution for simulating pedigrees as genetic databases continue to expand.V. Closing

[0302] The present disclosure is not to be limited in terms of the particular embodiments described in this application, which are intended as illustrations of various aspects. Many modifications and variations can be made without departing from its scope, as will be apparent to those skilled in the art. Functionally equivalent methods and apparatuses within the scope of the disclosure, in addition to those described herein, will be apparent to those skilled in the art from the foregoing descriptions. Such modifications and variations are intended to fall within the scope of the appended claims.

[0303] The above detailed description describes various features and operations of the disclosed systems, devices, and methods with reference to the accompanying figures. The example embodiments described herein and in the figures are not meant to be limiting. Other embodiments can be utilized, and other changes can be made, without departing from the scope of the subject matter presented herein. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, separated, and designed in a wide variety of different configurations.

[0304] With respect to any or all of the message flow diagrams, scenarios, and flow charts in the figures and as discussed herein, each step, block, and / or communication can represent a processing of information and / or a transmission of information in accordance with example embodiments. Alternative embodiments are included within the scope of theseexample embodiments. In these alternative embodiments, for example, operations described as steps, blocks, transmissions, communications, requests, responses, and / or messages can be executed out of order from that shown or discussed, including substantially concurrently or in reverse order, depending on the functionality involved. Further, more or fewer blocks and / or operations can be used with any of the message flow diagrams, scenarios, and flow charts discussed herein, and these message flow diagrams, scenarios, and flow charts can be combined with one another, in part or in whole.

[0305] A step or block that represents a processing of information can correspond to circuitry that can be configured to perform the specific logical functions of a herein-described method or technique. Alternatively or additionally, a step or block that represents a processing of information can correspond to a module, a segment, or a portion of program code (including related data). The program code can include one or more instructions executable by a processor for implementing specific logical operations or actions in the method or technique. The program code and / or related data can be stored on any type of computer readable medium such as a storage device including RAM, a disk drive, a solid-state drive, or another storage medium.

[0306] The computer readable medium can also include non-transitory computer readable media such as non-transitory computer readable media that store data for short periods of time like register memory and processor cache. The non-transitory computer readable media can further include non-transitory computer readable media that store program code and / or data for longer periods of time. Thus, the non-transitory computer readable media may include secondary or persistent long-term storage, like ROM, optical or magnetic disks, solid-state drives, or compact disc read only memory (CD-ROM), for example. The non-transitory computer readable media can also be any other volatile or non-volatile storage systems. Anon- transitory computer readable medium can be considered a computer readable storage medium, for example, or a tangible storage device.

[0307] Moreover, a step or block that represents one or more information transmissions can correspond to information transmissions between software and / or hardware modules in the same physical device. However, other information transmissions can be between software modules and / or hardware modules in different physical devices.

[0308] The particular arrangements shown in the figures should not be viewed as limiting. It should be understood that other embodiments could include more or less of each element shown in a given figure. Further, some of the illustrated elements can be combined or omitted. Yet further, an example embodiment can include elements that are not illustrated in the figures.

[0309] While various aspects and embodiments have been disclosed herein, other aspects and embodiments will be apparent to those skilled in the art. The various aspects and embodiments disclosed herein are for purpose of illustration and are not intended to be limiting, with the true scope being indicated by the following claims.

Claims

CLAIMSWhat is claimed is:

1. A non-transitory computer-readable medium, having stored thereon program instructions that, upon execution by a computing system, cause the computing system to perform operations comprising: obtaining first genetic sequence data of a first individual and second genetic sequence data of a second individual, wherein the first individual and the second individual are selected conditional on them sharing at least a threshold amount of identical -by-descent (IBD) segments of genetic sequence data; based on the first genetic sequence data and the second genetic sequence data, determining that the first individual and the second individual share N IBD segments of genetic sequence data, wherein the N IBD segments have respective lengths; identifying a plurality of candidate shared ancestor sets for the first individual and the second individual, wherein each of the candidate shared ancestor sets includes a respective one or two possible shared ancestors with a respective first number of meioses between the first individual and the respective one or two possible shared ancestors and a respective second number of meioses between the second individual and the respective one or two possible shared ancestors; for each of the candidate shared ancestor sets, determining probabilities that the N IBD segments of the respective lengths would occur given that the first individual and the second individual share at least one IBD segment; and based on the probabilities, selecting one of the candidate shared ancestor sets as most- recent IBD-contributing shared ancestors of the first individual and the second individual.

2. The non-transitory computer-readable medium of claim 1, wherein selecting one of the candidate shared ancestor sets as the most-recent IBD-contributing shared ancestors is also based on the respective lengths of a set of the N IBD segments.

3. The non-transitory computer-readable medium of claim 1, wherein selecting one of the candidate shared ancestor sets as the most-recent IBD-contributing shared ancestors is also based on a total length of the N IBD segments.

4. The non-transitory computer-readable medium of claim 1, wherein selectingone of the candidate shared ancestor sets as the most-recent IBD-contributing shared ancestors is also based on ranges of a total length of the N IBD segments mapped to relationship degrees.

5. The non-transitory computer-readable medium of claim 1, wherein the threshold amount of IBD segments is one IBD segment.

6. The non-transitory computer-readable medium of claim 1, wherein N is between 1 and 60, inclusive.

7. The non-transitory computer-readable medium of claim 1, wherein the respective first number of meioses and the respective second number of meioses are different.

8. The non-transitory computer-readable medium of claim 1 , the operations further comprising: based on IBD segments shared by the first individual, the second individual, and the candidate shared ancestor that was selected, determining a likelihood of a disease for the first individual or the second individual, wherein the disease has a genetic basis in at least one of the IBD segments shared.

9. The non-transitory computer-readable medium of claim 1 , the operations further comprising: based on IBD segments shared by the first individual, the second individual, and the candidate shared ancestor that was selected, determining a time of divergence or merger of a genetically isolated population.

10. The non-transitory computer-readable medium of claim 1, the operations further comprising: providing, for display on a graphical user interface, a tree-based representation of: (i) individuals including the first individual, the second individual, and the candidate shared ancestor that was selected, and (ii) generational relationships therebetween, wherein nodes of the tree-based representation display or can be actuated to display identifying information, genetic markers, or IBD segment data relating to the individuals.

11. The non-transitory computer-readable medium of claim 1, further comprising:providing, for display on a graphical user interface, a timeline-based representation of: (i) individuals including the first individual, the second individual, and the candidate shared ancestor that was selected, and (ii) presence or length of IBD segments of shared by the individuals, wherein icons display or can be actuated to display identifying information, genetic markers, or IBD segment locations relating to the individuals.

12. A non-transitory computer-readable medium, having stored thereon program instructions that, upon execution by a computing system, cause the computing system to perform operations comprising: determining a relationship between a first individual and a second individual, wherein the relationship includes one or two possible shared ancestors of the first individual and the second individual, wherein a first number of meioses is between the first individual and the one or two possible shared ancestors, and wherein a second number of meioses is between the second individual and the one or two possible shared ancestors; determining that the first individual and the second individual share at least one identical-by-descent (IBD) segment of genetic sequence data; and probabilistically sampling a set of recombination breakpoints of each of the meioses sequentially, conditioned on any previously-sampled recombination breakpoints and conditioned on at least one IBD segment being shared between the first individual and the second individual, wherein the set of recombination breakpoints is accepted when it meets an acceptance criterion.

13. The non-transitory computer-readable medium of claim 12, wherein the one or two possible shared ancestors is one shared ancestor.

14. The non-transitory computer-readable medium of claim 12, wherein the one or two possible shared ancestors are two shared ancestors.

15. The non-transitory computer-readable medium of claim 12, wherein the first number of meioses and the second number of meioses are different.

16. The non-transitory computer-readable medium of claim 12, wherein the recombination breakpoints comprise locations on chromosomes where breaks and subsequent recombinations occur during the meioses.

17. The non-transitory computer-readable medium of claim 12, wherein probabilistically sampling the set of recombination breakpoints comprises Bernoulli sampling of a parental haplotype on which copying for a meiosis starts.

18. The non-transitory computer-readable medium of claim 12, wherein probabilistically sampling the set of recombination breakpoints comprises Poisson sampling of a number of the recombination breakpoints of a meiosis.

19. The non-transitory computer-readable medium of claim 12, wherein probabilistically sampling the set of recombination breakpoints comprises the recombination breakpoints of a meiosis being sampled uniformly along a chromosome.

20. The non-transitory computer-readable medium of claim 12, wherein the acceptance criterion comprises the set of recombination breakpoints including IBD segments of a meiosis overlapping by a threshold length for the first individual and the second individual.

21. The non-transitory computer-readable medium of claim 12, the operations further comprising: based on the set of recombination breakpoints, deriving IBD sharing distributions for relationship estimation conditioned on shared IBD segments.

22. The non-transitory computer-readable medium of claim 12, the operations further comprising: based on the set of recombination breakpoints, deriving a pedigree for a plurality of individuals and relationships therebetween conditioned on shared IBD segments.

23. The non-transitory computer-readable medium of claim 22, wherein the plurality of individuals includes 20 or more individuals, and wherein at least some of the plurality of individuals are separated by 10 or more degrees of relatedness.

24. The non-transitory computer-readable medium of claim 12, further comprising: providing, for display on a graphical user interface, an interactive family tree that visually represents relationships between individuals including the relationship between thefirst individual and the second individual, wherein nodes in the interactive family tree represent each individual, wherein connecting lines indicate meioses from the first individual and the second individual to the one or two possible shared ancestors, and wherein coded markers represent presence of shared IBD segments.

25. The non-transitory computer-readable medium of claim 12, further comprising: providing, for display on a graphical user interface, a split-panel display, wherein one panel provides a pedigree chart representing familial connections between individuals including the relationship between the first individual and the second individual and their shared IBD segments, and wherein an adjacent panel provides a sequential, timeline-style visualization that represents recombination events across meioses.

26. A non-transitory computer-readable medium, having stored thereon program instructions that, upon execution by a computing system, cause the computing system to perform operations comprising: determining a pedigree including a set of relationships among a set of individuals, wherein each of the relationships has a respective one or two possible shared ancestors with a respective first number of meioses between a respective first individual and the respective one or two possible shared ancestors, and wherein each of the relationships also has a respective second number of meioses between a respective second individual and the respective one or two possible shared ancestors; determining which of the relationships are of individuals that share at least one identical-by-descent (IBD) segment of genetic sequence data; and for each of the relationships, probabilistically sampling a set of recombination breakpoints of each of the meioses sequentially, conditioned on any previously-sampled recombination breakpoints and conditioned on at least one IBD segment being shared between individuals of the relationship, wherein the set of recombination breakpoints is accepted when it meets an acceptance criterion.

27. The non-transitory computer-readable medium of claim 26, wherein the respective one or two possible shared ancestors is one shared ancestor.

28. The non-transitory computer-readable medium of claim 26, wherein the respective one or two possible shared ancestors are two shared ancestors.

29. The non-transitory computer-readable medium of claim 26, wherein the respective first number of meioses and the respective second number of meioses are different.

30. The non-transitory computer-readable medium of claim 26, wherein the recombination breakpoints comprise locations on chromosomes where breaks and subsequent recombinations occur during the meioses.

31. The non-transitory computer-readable medium of claim 26, wherein probabilistically sampling the set of recombination breakpoints comprises Bernoulli sampling of a parental haplotype on which copying for a meiosis starts.

32. The non-transitory computer-readable medium of claim 26, wherein probabilistically sampling the set of recombination breakpoints comprises Poisson sampling of a number of the recombination breakpoints of a meiosis.

33. The non-transitory computer-readable medium of claim 26, wherein probabilistically sampling the set of recombination breakpoints comprises the recombination breakpoints of a meiosis being sampled uniformly along a chromosome.

34. The non-transitory computer-readable medium of claim 26, wherein the acceptance criterion comprises the set of recombination breakpoints including IBD segments of a meiosis overlapping by a threshold length for the respective first individual and the respective second individual.

35. The non-transitory computer-readable medium of claim 26, the operations further comprising: based on the set of recombination breakpoints, deriving IBD sharing distributions for relationship estimation conditioned on shared IBD segments.

36. The non-transitory computer-readable medium of claim 26, the operations further comprising: based on the set of recombination breakpoints, deriving a pedigree for at least some of the individuals and relationships therebetween conditioned on shared IBD segments.

37. The non-transitory computer-readable medium of claim 36, wherein the pedigree includes 20 or more of the individuals, and wherein at least some of the individuals in the pedigree are separated by 10 or more degrees of relatedness.

38. The non-transitory computer-readable medium of claim 26, further comprising: providing, for display on a graphical user interface, an interactive family tree that visually represents relationships between the individuals, wherein nodes in the interactive family tree represent each of the individuals, wherein connecting lines indicate meioses from a pair of the individuals to the one or two possible shared ancestors, and wherein coded markers represent presence of shared IBD segments.

39. The non-transitory computer-readable medium of claim 26, further comprising: providing, for display on a graphical user interface, a split-panel display, wherein one panel provides a pedigree chart representing familial connections between the individuals and their shared IBD segments, and wherein an adjacent panel provides a sequential, timeline-style visualization that represents recombination events across meioses.

40. A non-transitory computer-readable medium, having stored thereon program instructions that, upon execution by a computing system, cause the computing system to perform operations comprising: obtaining first genetic sequence data of a first individual and second genetic sequence data of a second individual, wherein the first individual and the second individual are selected conditional on the first individual and the second individual sharing at least a threshold amount of identical-by-descent (IBD) segments of genetic sequence data; based on the first genetic sequence data and the second genetic sequence data, determining that the first individual and the second individual share N IBD segments of genetic sequence data, wherein the N IBD segments have respective lengths; and based on the N IBD segments of genetic sequence data and for a plurality of ancestral generations of the first individual and the second individual, estimating numbers of distinct common ancestors of the first individual and the second individual for each of the ancestral generations.

41. The non-transitory computer-readable medium of claim 40, the operationsfurther comprising: based on the numbers of distinct common ancestors as estimated, selecting one or more of the distinct common ancestors as most likely ancestors that contributed observable IBD segments to the first individual and the second individual.

42. The non-transitory computer-readable medium of claim 40, wherein estimating the numbers of distinct common ancestors of the first individual and the second individual for each of the ancestral generations is based on a recurrence relationship involving: an expected number of common ancestors between the first individual and the second individual in a given generation, and a number of distinct ancestors of the first individual in the given generation that are not ancestors of the second individual.

43. The non-transitory computer-readable medium of claim 40, wherein the threshold amount of IBD segments is one IBD segment.

44. The non-transitory computer-readable medium of claim 40, wherein N is between 1 and 60, inclusive.

45. The non-transitory computer-readable medium of claim 40, the operations further comprising: based on the N IBD segments shared by the first individual and the second individual, determining a likelihood of a disease for the first individual or the second individual, wherein the disease has a genetic basis in at least one of the N IBD segments.

46. The non-transitory computer-readable medium of claim 40, the operations further comprising: based on the N IBD segments shared by the first individual and the second individual, determining a time of divergence or merger of a genetically isolated population.

47. The non-transitory computer-readable medium of claim 40, the operations further comprising: providing, for display on a graphical user interface, a bar chart, wherein each bar along a horizontal axis represents a single ancestral generation, progressing chronologically into the past, and wherein a vertical axis represents an estimated count of the distinct common ancestorsin each of the ancestral generations.

48. The non-transitory computer-readable medium of claim 40, the operations further comprising: providing, for display on a graphical user interface, a heatmap, wherein rows of the heatmap represent generations progressing chronologically into the past, and wherein columns categorize the distinct common ancestors based on relatedness metrics or genetic contribution proportions.

49. A computer-implemented method configured to perform the operations of any of claims 1-48.

50. A computing system comprising: one or more processors; memory; and program instructions, stored in the memory, that upon execution by the one or more processors cause the computing system to perform the operations of any of claims 1-48.

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