Estimating device, estimating method, and estimating program

Abstract

Proposed is a new posterior distribution estimation technique. This estimating device comprises at least one memory and at least one processor, wherein, when estimating a posterior distribution of a state at a second time point based on observed values up to the second time point, in a case of a non-linear state space model in which the state distribution at the second time point, evolved over time on the basis of the state at a first time point, is described by a linear Gaussian distribution, and the distribution of observed values at the second time point, based on the state at the second time point, is described by a non-linear Gaussian distribution, the at least one processor uses a model that estimates a linear Gaussian distribution and that has parameters that are optimized to maximize a lower bound on a log-likelihood, to calculate a posterior distribution of the state at the second time point based on the observed values up to the second time point.

Description

Estimation device, estimation method, and estimation program

[0001] The present disclosure is the "Research and development of elemental technologies for providing user-optimized data using remote sensing technology" of the Ministry of Internal Affairs and Communications in fiscal years 2023 and 2024, and relates to an estimation device, an estimation method, and an estimation program.

[0002] As a posterior distribution estimation technique that uses a state space model to estimate the posterior distribution of a state at time t based on observed values ​​up to time t, for example, a particle filter is known. This technique makes it possible to estimate the posterior distribution even for a nonlinear state space model.

[0003] On the other hand, the above technique has a problem in that if the number of dimensions of the observations or the number of dimensions of the state of the state space model is large, the calculation cost becomes high.

[0004] Chopin, N. and Papaspiliopoulos, O. An introduction to Sequential Monte Carlo. Springer series in statistics. Springer, 2020.<URL: https: / / ci.nii.ac.jp / ncid / BC03234800>Daum, F. and Huang, J. Nonlinear filters with log-homotopy. In Drummond, OE and Teichgraeber, RD (eds.), Signal and Data Processing of Small Targets 2007, volume 6699, pp. 669918. International Society for Optics and Photonics, SPIE, 2007. <URL: https: / / doi.org / 10.1117 / 12.725684> Hu, C.-C. and van Leeuwen, PJ A particle flow filter for high-dimensional system applications. Quarterly Journal of the Royal Meteorological Society, 147(737):2352-2374, 2021. <URL: https: / / doi.org / 10.1002 / qj.4028>

[0005] This disclosure proposes a new posterior distribution estimation technique.

[0006] An estimation device according to one aspect of the present disclosure has, for example, the following configuration: That is, the estimation device includes at least one memory and at least one processor, wherein, in a case of a nonlinear state space model in which a distribution of a state at a second time point evolved over time based on a state at a first time point is described by a linear Gaussian distribution and a distribution of observation values ​​at the second time point is described by a nonlinear Gaussian distribution based on the state at the second time point, when estimating a posterior distribution of the state at the second time point based on observation values ​​up to the second time point, the at least one processor calculates the posterior distribution of the state at the second time point based on the observation values ​​up to the second time point using a model that estimates a linear Gaussian distribution, the model having parameters optimized to maximize a lower bound of log-likelihood.

[0007] FIG. 1 is a diagram illustrating an example of a system configuration in the parameter optimization phase of an estimation system according to a first embodiment. FIG. 2 is a diagram illustrating an example of the hardware configuration of an information processing device. FIG. 3 is a diagram illustrating a specific example of processing by a state space model input unit. FIG. 4 is a diagram illustrating a specific example of processing by a prediction and update step generation unit and a specific example of processing by a simulation unit. FIG. 5 is a first diagram illustrating a specific example of processing by a dynamics model linearization unit. FIG. 6 is a first diagram illustrating a specific example of processing by a prediction and update step transformation unit. FIG. 7 is a first diagram illustrating a specific example of processing by a DBF generation unit. FIG. 8 is a first diagram illustrating a specific example of processing by an ELBO generation unit. FIG. 9 is a first diagram illustrating a specific example of processing by a parameter optimization unit. FIG. 10 is an example of a first flowchart illustrating the flow of parameter optimization processing. FIG. 11 is a diagram illustrating an example of a system configuration in the estimation phase of an estimation system according to a first embodiment. FIG. 12 is a diagram illustrating an example of the hardware configuration of a server device. FIG. 13 is a first diagram illustrating a specific example of processing by a posterior distribution estimation unit. FIG. 14 is an example of a first flowchart showing the flow of estimation processing. FIG. 15 is a diagram showing an example of a system configuration in the parameter optimization phase of an estimation system according to a second embodiment. FIG. 16 is a first diagram showing a specific example of processing by the dynamics model linearizer. FIG. 17 is a second diagram showing a specific example of processing by the prediction and update step transformer. FIG. 18 is a second diagram showing a specific example of processing by the DBF generator. FIG. 19 is a second diagram showing a specific example of processing by the ELBO generator. FIG. 20 is a second diagram showing a specific example of processing by the parameter optimizer. FIG. 21 is an example of a second flowchart showing the flow of parameter optimization processing. FIG. 22 is a diagram showing an example of a system configuration in the estimation phase of an estimation system according to the second embodiment. FIG. 23 is a second diagram showing a specific example of processing by the posterior distribution estimator. FIG. 24 is an example of a second flowchart showing the flow of estimation processing. FIG. 25 is a second diagram showing a specific example of processing by the dynamics model linearizer. FIG. 26 is a diagram illustrating an example of a system configuration in the parameter optimization phase of the estimation system according to the third embodiment.FIG. 27A is a third diagram showing a specific example of processing by the prediction and update step transformation unit. FIG. 27B is a third diagram showing a specific example of processing by the DBF generation unit. FIG. 27C is a third diagram showing a specific example of processing by the ELBO generation unit. FIG. 28 is a third diagram showing a specific example of processing by the parameter optimization unit. FIG. 29 is an example of a third flowchart showing the flow of parameter optimization processing. FIG. 30 is a diagram showing an example of a system configuration in the parameter optimization phase of an estimation system according to a fourth embodiment. FIG. 31 is a diagram showing a specific example of processing by the prediction and update step generation unit. FIG. 32 is a diagram showing an example of a system configuration in the parameter optimization phase of an estimation system according to a fifth embodiment. FIG. 33 is a diagram showing a specific example of processing by the smoothing distribution generation unit. FIG. 34A is a first diagram showing a specific example of processing by the ELBO generation unit. FIG. 34B is a second diagram showing a specific example of processing by the ELBO generation unit. FIG. 35 is a first diagram showing a specific example of processing by the calculability condition input unit. FIG. 36 is a fourth diagram showing a specific example of processing by the parameter optimization unit. FIG. 37 is a diagram showing a specific example of processing by the filtering distribution generation unit. FIG. 38 is a diagram showing a specific example of processing by the KL distance generation unit. FIG. 39 is a second diagram showing a specific example of processing by the calculability condition input unit. FIG. 40 is a fifth diagram showing a specific example of processing by the parameter optimization unit. FIG. 41 is an example of a fourth flowchart showing the flow of parameter optimization processing. FIG. 42 is a diagram showing an example of a system configuration in the estimation phase of the estimation system according to the fifth embodiment. FIG. 43 is a third diagram showing a specific example of processing by the posterior distribution estimation unit. FIG. 44 is an example of a third flowchart showing the flow of estimation processing. FIG. 45 is a diagram showing an example of a system configuration in the parameter optimization phase of the raindrop particle diameter distribution estimation system. FIG. 46 is a diagram showing an example of a system configuration in the estimation phase of the raindrop particle diameter distribution estimation system.

[0008] Hereinafter, each embodiment will be described with reference to the accompanying drawings. In this specification and drawings, components having substantially the same functional configurations are designated by the same reference numerals, and redundant description will be omitted.

[0009] [First Embodiment] <System Configuration of Estimation System (Parameter Optimization Phase)> First, a system configuration in the parameter optimization phase of an estimation system according to the first embodiment will be described. The estimation system according to the first embodiment is a system that estimates a state in a physical space using a state space model. A "state space model" is a model that assumes the existence of a state in a physical space that cannot be directly observed, and is composed of a "dynamics model" and an "observation model."

[0010] A "dynamics model" is a model that estimates the state at time t (second time point) after time evolution based on the state at time t-1 (first time point). An "observation model" is a model that estimates the distribution of observed values ​​at time t (second time point) based on the state at time t (second time point).

[0011] The estimation system according to the first embodiment recursively calculates a "prediction step" including a dynamics model and an "update step" including an observation model, thereby estimating the posterior distribution of the state at time t from the observed values ​​up to time t.

[0012] 1 is a diagram showing an example of a system configuration in the parameter optimization phase of an estimation system according to Embodiment 1. As shown in Fig. 1, the estimation system 100 according to Embodiment 1 includes a server device 110 and an information processing device 120 in the parameter optimization phase.

[0013] A parameter optimization program is installed in the information processing device 120. By executing the program, the information processing device 120 functions as a state space model input unit 121, a prediction and update step generation unit 122, a dynamics model linearization unit 123, a prediction and update step transformation unit 124, a DBF generation unit 125, and an ELBO generation unit 126.

[0014] The state space model input unit 121 inputs a state space model used to estimate a state in physical space. In the first embodiment, a dynamics model constituting the state space model is described by a nonlinear non-Gaussian distribution in which the operator is nonlinear and the process noise does not follow a Gaussian distribution. Also in the first embodiment, an observation model constituting the state space model is described by a nonlinear Gaussian distribution in which the operator is nonlinear and the observation noise follows a Gaussian distribution. In other words, in the first embodiment, the state space model is a nonlinear state space model. Note that in the first embodiment, it is assumed that the parameters of the dynamics model and the parameters of the observation model are known.

[0015] The prediction and update step generator 122 generates prediction steps including a dynamics model and update steps including an observation model. The prediction steps generated by the prediction and update step generator 122 are described by a nonlinear non-Gaussian distribution, and the update steps generated by the prediction and update step generator 122 are described by a nonlinear Gaussian distribution. The prediction steps and update steps generated by the prediction and update step generator 122 are input to the server device 110 and used to calculate simulation values.

[0016] The dynamics model linearization unit 123 linearizes the dynamics model described by a nonlinear non-Gaussian distribution by introducing a latent variable into the dynamics model, and describes the dynamics model by a linear Gaussian distribution.

[0017] The prediction and update step modification unit 124 describes the prediction step with a linear Gaussian distribution by introducing a dynamics model described with a linear Gaussian distribution into the prediction step of the prediction steps and update steps generated by the prediction and update step generation unit 122. This makes it possible to analytically calculate the prediction step.

[0018] The DBF generation unit 125 generates an update step including an inverse observation operator, thereby making the update step calculable. Furthermore, the DBF generation unit 125 generates a DBF (Deep Bayesian Filter) that estimates a posterior distribution by recursively calculating a prediction step and an update step including an inverse observation operator.

[0019] The ELBO generator 126 generates an evidence lower bound (ELBO) to optimize the parameters included in the DBF. The ELBO is a function that indicates the lower bound of the log-likelihood. The ELBO generated by the ELBO generator 126 is input to the server device 110 and used to optimize the parameters included in the DBF.

[0020] A calculation program is installed in the server device 110 , and by executing the program, the server device 110 functions as a simulation unit 111 and a parameter optimization unit 112 .

[0021] The simulation unit 111 acquires the prediction step and update step generated by the information processing device 120 and executes a simulation. As a result, the simulation unit 111 obtains the simulation value (state (z ｔ ) and the observed value (o ｔ The simulation value at each time point t generated by the simulation unit 111 is stored in the simulation value storage unit 114.

[0022] The parameter optimization unit 112 optimizes the parameters included in the DBF using the simulation values ​​at each time point t stored in the simulation value storage unit 114. Specifically, the parameter optimization unit 112 has an ELBO maximization unit 113, and optimizes the parameters included in the DBF so as to maximize the ELBO (lower bound of the log-likelihood) generated by the ELBO generation unit 126. By maximizing the ELBO, the KL distance (Kullback-Leibler Divergence) between the true posterior distribution and the DBF is minimized. In other words, the ELBO maximization unit 113 optimizes the parameters included in the DBF so as to minimize the KL distance between the true posterior distribution and the DBF.

[0023] <Hardware configuration of server device and information processing device> Next, a description will be given of the hardware configuration of the server device 110 and the information processing device 120. Note that the server device 110 and the information processing device 120 have similar hardware configurations, and therefore the hardware configuration of the information processing device 120 will be described here.

[0024] 2 is a diagram showing an example of the hardware configuration of an information processing device. The information processing device 120 has, as its components, a processor 201, a main storage device 202 (memory), an auxiliary storage device 203 (memory), a network interface 204, and a device interface 205. The information processing device 120 may be realized as a computer in which these components are connected via a bus 206. Note that, in the example of FIG. 2, the information processing device 120 is shown as having one of each component, but the information processing device 120 may also have multiple of the same component.

[0025] Various computations of the information processing device 120 may be executed in parallel using one or more processors. Furthermore, various computations may be distributed to multiple computing cores within the processor 201 and executed in parallel. Furthermore, some or all of the processes, means, etc. disclosed herein may be executed by an external device 230 (at least one of a processor and a storage device) provided on a cloud that can communicate with the information processing device 120 via the network interface 204.

[0026] The processor 201 may be an electronic circuit (processing circuit, processing circuitry, CPU, GPU, FPGA, ASIC, etc.). The processor 201 may also be a semiconductor device including a dedicated processing circuit. The processor 201 is not limited to an electronic circuit using electronic logic elements, and may be realized by an optical circuit using optical logic elements. The processor 201 may also include an arithmetic function based on quantum computing.

[0027] The processor 201 performs various calculations based on various data and commands input from each device, etc., in the internal configuration of the information processing device 120, and outputs the calculation results and control signals to each device, etc. The processor 201 controls each component included in the information processing device 120 by executing an OS (Operating System), applications, etc.

[0028] Furthermore, processor 201 may refer to one or more electronic circuits arranged on a single chip, or may refer to one or more electronic circuits arranged on two or more chips or devices. When multiple electronic circuits are used, the respective electronic circuits may communicate with each other via wires or wirelessly.

[0029] The main memory device 202 is a memory device that stores instructions executed by the processor 201 and various data, and the various data stored in the main memory device 202 is read by the processor 201. The auxiliary memory device 203 is a memory device other than the main memory device 202. Note that these memory devices refer to any electronic component that can store various data, and may be semiconductor memory. The semiconductor memory may be either volatile memory or non-volatile memory. The memory device for saving various data in the information processing device 120 may be realized by the main memory device 202 or the auxiliary memory device 203, or may be realized by an internal memory built into the processor 201.

[0030] Furthermore, multiple processors 201 may be connected (coupled) to one main storage device 202, or a single processor 201 may be connected. Alternatively, multiple main storage devices 202 may be connected (coupled) to one processor 201. When the information processing device 120 is configured with at least one main storage device 202 and multiple processors 201 connected (coupled) to this at least one main storage device 202, it may include a configuration in which at least one processor of the multiple processors 201 is connected (coupled) to at least one main storage device 202.

[0031] The network interface 204 is an interface for connecting to a communication network 220 wirelessly or via a wired connection.

[0032] The device interface 205 is an interface such as a USB that directly connects to an external device 240 .

[0033] The external device 240 may be, for example, an input device. In this embodiment, the input device is, for example, an electronic device such as a camera, a microphone, various sensors, a keyboard, a mouse, or a touch panel, and provides acquired information to the information processing device 120.

[0034] Furthermore, the external device 240 may be, for example, an output device. In this embodiment, the output device may be, for example, a display device such as an LCD (Liquid Crystal Display), a CRT (Cathode Ray Tube), a PDP (Plasma Display Panel), or an organic EL (Electro Luminescence) panel, or may be a speaker that outputs sound or the like.

[0035] The external device 240 may also be a storage device (memory). For example, the external device 240 may be a network storage or the like, or may be a storage such as an HDD.

[0036] Furthermore, the external device 240 may be a device that has some of the functions of the components of the information processing device 120. In other words, the information processing device 120 may transmit and receive processing results to and from the external device 240.

[0037] <Specific Example of Processing by Each Unit of Estimation System (Parameter Optimization Phase)> Next, a specific example of processing by each unit of the server device 110 and the information processing device 120 included in the estimation system 100 in the parameter optimization phase will be described.

[0038] (1) Specific Example of Processing by State Space Model Input Unit 121 First, a specific example of processing by the state space model input unit 121 of the information processing device 120 will be described. Fig. 3 is a diagram showing a specific example of processing by a state space model. As shown in Fig. 3, the state space model input unit 121 inputs a dynamics model 310 and an observation model 320 that constitute the state space model.

[0039] The dynamics model 310 calculates the state (z ｔ－１ ) based on the time-evolved state at time t (z ｔ ) is described by a nonlinear non-Gaussian distribution model p(z ｔ |z ｔ－１ )

[0040] The observation model 320 is the state (z ｔ ) based on the observation value (o ｔ ) is estimated by a nonlinear Gaussian model p(o ｔ |z ｔ )

[0041] (2) Specific Examples of Processing by the Prediction and Update Step Generator 122 and the Simulation Unit 111 Next, a specific example of processing by the prediction and update step generator 122 of the information processing device 120 and a specific example of processing by the simulation unit 111 of the server device 110 will be described. Fig. 4 is a diagram showing a specific example of processing by the prediction and update step generator and a specific example of processing by the simulation unit. As shown in Fig. 4, the prediction and update step generator 122 generates prediction steps 410 and update steps 420.

[0042] The prediction step 410 predicts the observations (o １：ｔ－１ ) based on the state (z ｔ ) distribution estimation step p(z ｔ |o １：ｔ－１ Specifically, the prediction step 410 is performed by: １：ｔ－１ ) at time t-1 estimated based on ｔ－１ ) posterior distribution p(z ｔ－１ |o １：ｔ－１ )) and the result calculated by the dynamics model (state (z ｔ－１ ) at time t estimated based on ｔ ) distribution p(z ｔ |z ｔ－１ )) and . In FIG. 4, reference numeral 430 indicates a state space model with Markov properties. The state space model with Markov properties is, for example, a state (z ｔ ) is the state (z ｔ－１ ) and any state (z ｔ ) is independent of the observed value at time t (o ｔ ) is the state at time t (z ｔ ) and does not depend on the state at a past point in time (before point in time t).

[0043] The update step 420 is the process of updating the observations (o １：ｔ ) based on the distribution of the state (z ｔ Step p(z ｔ |o １：ｔ Specifically, the update step 420 is performed by: １：ｔ－１ ) at time t estimated based on ｔ ) distribution p(z ｔ |o １：ｔ－１)) and the result calculated by the observation model (state at time t (z ｔ ) based on the observed value (o ｔ ) distribution p(o ｔ |z ｔ )) and is calculated based on .

[0044] The prediction steps 410 and update steps 420 generated by the prediction and update step generation unit 122 are transmitted to the simulation unit 111 of the server device 110 .

[0045] As shown in FIG. 4, the simulation unit 111 acquires the prediction step 410 and the update step 420 generated by the prediction and update step generation unit 122, and executes a simulation. As a result, the simulation unit 111 obtains the simulation value (state (z ｔ ) and the observed value (o ｔ The simulation value at each time point t generated by the simulation unit 111 is stored in the simulation value storage unit 114.

[0046] In the example of FIG. 4, the simulation values ​​are (z ０ , o ０ ), (z １ , o １ ), ..., (z ｘ , o ｘ ) is stored in the simulation value storage unit 114.

[0047] (3) Specific Example of Processing by the Dynamics Model Linearization Unit 123 Next, a description will be given of a specific example of processing by the dynamics model linearization unit 123 of the information processing device 120. Fig. 5 is a first diagram showing a specific example of processing by the dynamics model linearization unit.

[0048] As shown in FIG. 5, the dynamics model linearization unit 123 introduces a latent variable h (reference numeral 510) and linearizes the dynamics model (reference numeral 520).

[0049] The latent variable h is a variable with linear dynamics. By introducing the latent variable h, the dynamics model p(zｔ |z ｔ－１ ) is p(h ｔ |h ｔ－１ ) can be written as the dynamics model p(h ｔ |h ｔ－１ ) is the latent variable (h ｔ－１ ) based on the latent variable (h ｔ ) is a model described by a linear Gaussian distribution that estimates the distribution of ｔ |h ｔ－１ ) includes A and Q as parameters.

[0050] (4) Specific Example of Processing by the Prediction and Update Step Transformation Unit 124 Next, a specific example of processing by the prediction and update step transformation unit 124 of the information processing device 120 will be described. Fig. 6 is a first diagram showing a specific example of processing by the prediction and update step transformation unit. As shown in Fig. 6, the prediction and update step transformation unit 124 transforms the prediction step 410 and the update step 420 by introducing the latent variable h, and generates a prediction step 610 and an update step 620.

[0051] The post-transformation prediction step 610 is the prediction of the observations (o １：ｔ－１ ) based on the latent variable (h ｔ Specifically, the distribution of p(h) is estimated in the post-transformation prediction step 610 (p(h ｔ |o １：ｔ－１ )) are the results calculated in the post-deformation update step 620 (the observed values ​​(o １：ｔ－１ ) and the latent variable (h ｔ－１ ) posterior distribution p(h ｔ－１ |o １：ｔ－１ )) and the results calculated by the dynamics model (latent variables (h ｔ－１ ) and the latent variable (h ｔ ) distribution p(h ｔ |h ｔ：ｔ－１)) and Note that the post-transformation prediction step 610 can be analytically calculated since it is described by a linear Gaussian distribution.

[0052] The post-transformation update step 620 is the observations (o １：ｔ ) based on the latent variable (h ｔ Specifically, the post-transformation update step 620 (p(h ｔ |o １：ｔ )) are the results calculated in the prediction step (observations (o １：ｔ－１ ) and the latent variable (h ｔ ) distribution p(h ｔ |o １：ｔ－１ )) and the results calculated by the observation model (latent variables (h ｔ ) based on the observed value (o ｔ ) distribution p(o ｔ |h ｔ )) and However, the post-deformation update step 620 cannot be calculated analytically because it is still described by a nonlinear Gaussian distribution.

[0053] (5) Specific Example of Processing by DBF Generator 125 Next, a specific example of processing by the DBF generator 125 of the information processing device 120 will be described. FIG. 7 is a first diagram illustrating a specific example of processing by the DBF generator. As shown in FIG. 7, the DBF generator 125 introduces an inverse observation operator (reference numeral 710), makes an update step including the inverse observation operator computable, and generates a DBF (reference numeral 720). The DBF generator 125 also defines an operator that returns the posterior distribution of the latent variables to a state in physical space (reference numeral 730). Here, the content of the processing by the DBF generator 125 will be described with reference to reference numeral 700 (a diagram showing the relationship between each variable and parameter in a DBF).

[0054] The inverse observation operator r(h ｔ |o ｔ ) is an operator that estimates the distribution of latent variables at time t based on the observed values ​​at time t (reference numeral 700). As shown in reference numeral 710, the inverse observation operator r(hｔ |o ｔ ) includes the parameter of the operator f and the parameter of the operator G as parameters.

[0055] As shown at 720, the DBF is the sum of the observations (o １：ｔ ) based on the latent variable (h ｔ ) calculated in the prediction step (observations (o) from time 1 to time t-1) １：ｔ－１ ) and the latent variable (h ｔ ) distribution p(h ｔ |o １：ｔ－１ )) and the inverse observation operator r(h ｔ |o ｔ ) (observation value at time t (o ｔ ) and the latent variable (h ｔ ) distribution r(h ｔ |o ｔ )) and the latent variable (h ｔ ) is calculated.

[0056] The operator that returns the posterior distribution of the latent variable to the state of the physical space is an operator that estimates the probability distribution of the state of the physical space at time t under the latent variable at time t. As shown by the reference numeral 730, the operator p(z ｔ |h ｔ ) includes the parameter of the operator φ and the parameter R as parameters.

[0057] (6) Specific Example of Processing by the ELBO Generator 126 Next, a specific example of processing by the ELBO generator 126 of the information processing device 120 will be described. FIG. 8 is a first diagram showing a specific example of processing by the ELBO generator. As shown in FIG. 8, the ELBO generator 126 generates a simulation value (state (z ｔ ) and the observed value (o ｔ ) is generated (reference numeral 830).

[0058] As shown at 830, the ELBO includes an approximate model q(hｔ |o １：ｔ ) and the operator p(z ｔ |h ｔ ) is included (reference numeral 730). ｔ |o １：ｔ ) is a DBF (reference numeral 720). As described above, a DBF is a model that estimates a posterior distribution by recursively calculating a prediction step and an update step including an inverse observation operator.

[0059] The ELBO generated by the ELBO generating unit 126 is input to the server device 110 and used to optimize the parameters included in the DBF and the parameters included in the operator that returns to the state of the physical space.

[0060] As mentioned above, maximizing the ELBO minimizes the KL distance between the true posterior distribution and the DBF.

[0061] (7) Specific Example of Processing by Parameter Optimization Unit 112 Next, a specific example of processing by the parameter optimization unit 112 of the server device 110 will be described. Fig. 9 is a first diagram showing a specific example of processing by the parameter optimization unit. As shown in Fig. 9, the parameter optimization unit 112 has an ELBO maximization unit 113, and optimizes parameters included in the DBF and parameters included in the operator for restoring to the state of physical space so as to maximize the ELBO generated by the ELBO generation unit 126.

[0062] Specifically, the ELBO maximization unit 113 maximizes the likelihood of the simulation value by using the parameters included in the DBF and the parameters included in the operator that returns the state to the physical space. ｔ |o ｔ ) parameters (parameters of operator f, parameters of operator G), linearized dynamics model p(h ｔ |h ｔ－１ ) parameters (parameter A, parameter Q), the state of the physical space (z ｔ ) ｔ |h ｔ ) parameters (parameters of operator φ, parameter R), are optimized.

[0063] <Flow of Parameter Optimization Processing> Next, a description will be given of the flow of parameter optimization processing in the parameter optimization phase of the estimation system 100 according to the first embodiment. Fig. 10 is an example of a first flowchart showing the flow of the parameter optimization processing.

[0064] In step S1001, the information processing device 120 receives an input of a state space model used to estimate a state in a physical space.

[0065] In step S1002, the information processing device 120 generates a prediction step including a dynamics model and an update step including an observation model.

[0066] In step S1003, the server device 110 acquires the prediction steps and update steps generated by the information processing device 120 and executes a simulation.

[0067] In step S1004 , the server device 110 stores the generated simulation values ​​at each time point t in the simulation value storage unit 114 .

[0068] In step S1005, the information processing device 120 linearizes the dynamics model described by the nonlinear non-Gaussian distribution by introducing a latent variable, and describes the dynamics model by a linear Gaussian distribution.

[0069] In step S1006, the information processing device 120 describes the prediction step using a linear Gaussian distribution by introducing a dynamics model described using a linear Gaussian distribution into the prediction step.

[0070] In step S1007, the information processing apparatus 120 generates a DBF.

[0071] In step S1008, the information processing device 120 generates an ELBO for optimizing the parameters included in the DBF and the parameters included in the operator for restoring the state of the physical space.

[0072] In step S1009, the server device 110 inputs the simulation values ​​stored in step S1004, and optimizes the parameters included in the DBF and the parameters included in the operator that returns to the state of physical space so as to maximize ELBO.

[0073] <System Configuration of Estimation System (Estimation Phase)> Next, a system configuration of the estimation system according to the first embodiment in the estimation phase will be described. Fig. 11 is a diagram showing an example of the system configuration of the estimation system according to the first embodiment in the estimation phase. As shown in Fig. 11, the estimation system 1100 according to the first embodiment includes a measurement device 1140 and a server device 1110 (an example of an estimation device) in the estimation phase.

[0074] The measurement device 1140 is a device that measures the physical space 1130, and for example, the observed value o １ ~o ｔ Output.

[0075] An estimation program is installed in the server device 1110 , and when the program is executed, the server device 1110 functions as a posterior distribution estimation unit 1120 .

[0076] The posterior distribution estimation unit 1120 has a model in which parameters (parameters of the operator f, parameters of the operator G, parameters A, and parameters Q) are optimized. １ ~o ｔ Based on this, the latent variable (h ｔ ) posterior distribution p(h ｔ |o １：ｔ ) is calculated.

[0077] The posterior distribution estimation unit 1120 also calculates the latent variable (h ｔ ) under the state of the physical space at time t (z ｔ The posterior distribution estimation unit 1120 estimates the distribution of the state of the physical space (z ｔ ) distribution sample, the posterior distribution p(z ｔ |o １：ｔ) sample (z ｔ＿ｓａｍｐｌｅ ) is calculated.

[0078] <Hardware Configuration of Server Device> Next, the hardware configuration of the server device 1110 will be described. Fig. 12 is a diagram showing an example of the hardware configuration of the server device. The server device 1110 has, as components, a processor 1201, a main storage device 1202 (memory), an auxiliary storage device 1203 (memory), a network interface 1204, and a device interface 1205. The server device 1110 may be realized as a computer in which these components are connected via a bus 1206. Note that the components of the server device 1110 shown in Fig. 12 are the same as the components of the information processing device 120 shown in Fig. 2, and therefore description of each component will be omitted here.

[0079] <Specific Example of Processing by the Posterior Distribution Estimation Unit of the Estimation System (Estimation Phase)> Next, a specific example of processing by the posterior distribution estimation unit 1120 of the server device 1110 included in the estimation system 1100 in the inference phase will be described. FIG. 13 is a first diagram showing a specific example of processing by the posterior distribution estimation unit. As shown in FIG. 13, the posterior distribution estimation unit 1120 estimates the observed values ​​(o １：ｔ ) based on the latent variable (h ｔ ) posterior distribution p(h ｔ |o １：ｔ Specifically, the posterior distribution estimation unit 1120 estimates the observed value o １ ~o ｔ The model shown by the reference numeral 1121 is executed by inputting the following. The model shown by the reference numeral 1121 is an inverse observation operator r(h ｔ |o ｔ ) and the observed values ​​from time 1 to time t-1 (o １：ｔ－１ ) based on the latent variable (h ｔ ) distribution p(h ｔ |o １：ｔ－１ ) and the latent variable (h ｔ) posterior distribution p(h ｔ |o １：ｔ ) is calculated.

[0080] The posterior distribution estimation unit 1120 also calculates the latent variable (h ｔ ) under the condition that the state of the physical space at time t (z ｔ The operator 1122 estimates the distribution of the latent variable (h ｔ ) under the state of the physical space at time t (z ｔ ) at time t. ｔ ) from the distribution sample of the state of the physical space at time t (z ｔ ) to calculate a sample of the posterior distribution.

[0081] <Flow of Estimation Process> Next, a description will be given of the flow of estimation process in the estimation phase of the estimation system 1100 according to the first embodiment. Fig. 14 is an example of a first flowchart showing the flow of the estimation process.

[0082] In step S1401, the server device 1110 acquires the observed values ​​measured by the measurement device 1140 from time 1 to time t.

[0083] In step S1402, the server device 1110 estimates the posterior distribution of the latent variables at time t based on the observed values ​​from time 1 to time t.

[0084] In step S1403, the server device 1110 calculates a sample of the posterior distribution of the state of the physical space at time t from a sample of the distribution of the latent variables at time t.

[0085] In step S1404, the server device 1110 outputs the posterior distribution of the latent variables at time t based on the observed values ​​from time 1 to time t, and a sample of the posterior distribution of the state of the physical space at time t.

[0086] In step S1405, the server device 1110 determines whether or not to continue the estimation process. If it is determined in step S1405 that the estimation process should be continued (YES in step S1405), the process returns to step S1401. In this case, the server device 1110 advances the time point t and then executes the processes of steps S1401 to S1405. On the other hand, if it is determined in step S1405 that the estimation process should not be continued (NO in step S1405), the estimation process ends.

[0087] <Summary> As is clear from the above description, the estimation system 1100 according to the first embodiment calculates, in a state space model, a distribution p(z ｔ |z ｔ－１ ) is the latent variable (h ｔ ) is described by a linear Gaussian distribution, and the distribution p(o) of the observation value at time t is determined based on the state at time t. ｔ |z ｔ ) is described by a nonlinear Gaussian distribution, when estimating the posterior distribution of the state at time t based on the observed values ​​up to time t, the posterior distribution of the latent variable at time t based on the observed values ​​up to time t is calculated using a DBF that introduces an inverse observation operator and whose parameters are optimized to maximize the lower bound of the log-likelihood.

[0088] As described above, in the first embodiment, in a state space model, the following steps are performed: a latent variable is introduced to make the prediction step calculable; an inverse observation operator is introduced to make the update step calculable; and the prediction step and the update step are recursively calculated to generate a DBF that estimates the posterior distribution, thereby estimating the posterior distribution. Furthermore, in the first embodiment, parameters included in the DBF are optimized to maximize the lower bound of the log-likelihood. Then, in the first embodiment, the posterior distribution of the latent variable is calculated using the DBF with optimized parameters.

[0089] As a result, according to the first embodiment, it is possible to reduce the computational cost required for performing estimation with the same accuracy as that of conventional posterior distribution estimation techniques. In other words, according to the first embodiment, it is possible to propose a new posterior distribution estimation technique that is more computationally cost effective than conventional posterior distribution estimation techniques.

[0090] [Second Embodiment] In the first embodiment, the prediction step is made calculable by introducing a latent variable. However, the method for making the prediction step calculable is not limited to this. For example, the prediction step may be made calculable by linearizing the dynamics model using a Koopman operator. The second embodiment will be described below, focusing on the differences from the first embodiment.

[0091] <System Configuration of Estimation System (Parameter Optimization Phase)> First, a system configuration in the parameter optimization phase of an estimation system according to the second embodiment will be described. Fig. 2 is a diagram showing an example of the system configuration in the parameter optimization phase of the estimation system according to the second embodiment. The differences between the system configuration in the parameter optimization phase of the estimation system 100 according to the first embodiment, which was described in the first embodiment using Fig. 1 , are that the estimation system 1500 shown in Fig. 15 performs parameter optimization processing using observation values ​​obtained by measuring the physical space 1130 by the measurement device 1140, the functions of the dynamics model linearizer 1523, the prediction and update step transformer 1524, the DBF generator 1525, and the ELBO generator 1526 included in the information processing device 120 are different, and the function of the ELBO maximizer 1513 of the parameter optimizer 112 included in the server device 110 is different.

[0092] The dynamics model linearization unit 1523 converts the dynamics model described by a nonlinear Gaussian distribution using a Koopman operator, thereby describing the dynamics model by a linear Gaussian distribution.

[0093] The prediction and update step modification unit 1524 describes the prediction step using a linear Gaussian distribution by introducing a dynamics model described using a linear Gaussian distribution into the prediction step of the prediction steps and update steps generated by the prediction and update step generation unit 122. This makes it possible to analytically calculate the prediction step. Note that at this stage, the update step cannot be analytically calculated because it is still described using a nonlinear Gaussian distribution.

[0094] The DBF generator 1525 generates an update step including an inverse observation operator, thereby making the update step calculable. Furthermore, the DBF generator 1525 generates a DBF that estimates a posterior distribution by recursively calculating a prediction step and an update step including an inverse observation operator.

[0095] The ELBO generator 1526 generates an ELBO in order to optimize the parameters included in the DBF. The ELBO generated by the ELBO generator 1526 is input to the server device 110 and used to optimize the parameters included in the DBF.

[0096] The parameter optimization unit 112 optimizes the parameters included in the DBF using the observed values ​​at each time point t stored in the observed value storage unit 1514. Specifically, the parameter optimization unit 112 has an ELBO maximization unit 1513, and optimizes the parameters included in the DBF so as to maximize the ELBO (lower bound of the log-likelihood) generated by the ELBO generation unit 1526.

[0097] <Specific Example of Processing by Each Unit of Estimation System (Parameter Optimization Phase)> Next, a specific example of processing by each unit of the server device 110 included in the estimation system 1500 in the parameter optimization phase and a specific example of processing by each unit of the information processing device 120 will be described. However, here, a specific example of processing by the ELBO maximization unit 1513 of the parameter optimization unit 112, among the units of the server device 110, will be described. Also, here, a specific example of processing by the dynamics model linearization unit 1523, the prediction and update step deformation unit 1524, the DBF generation unit 1525, and the ELBO generation unit 1526, among the units of the information processing device 120, will be described.

[0098] (1) Specific Example of Processing by the Dynamics Model Linearization Unit 1523 First, a specific example of processing by the dynamics model linearization unit 1523 of the information processing device 120 will be described. FIG. 16 is a first diagram showing a specific example of processing by the dynamics model linearization unit. The difference from the dynamics model linearization unit 123 described with reference to FIG. 5 in the first embodiment is that the Koopman operator is used to perform variable transformation (x ｔ = φ(z ｔ ) and the dynamics model is linearized.

[0099] The dynamics model p(x ｔ |x ｔ－１ ) is a model described by a linear Gaussian distribution, and the state of the physical space at time t-1 (z ｔ－１ ) is transformed using the Koopman operator to obtain the state of the virtual space (x ｔ－１ ) based on the state of the virtual space after transformation at the time point t after time evolution (x ｔ ) is a model that estimates the distribution of the dynamics model p(x ｔ |x ｔ－１ ) includes a parameter A and a parameter Q (reference numeral 1610).

[0100] (2) Specific Example of Processing by the Prediction and Update Step Transformation Unit 1524 Next, a specific example of processing by the prediction and update step transformation unit 1524 of the information processing device 120 will be described. Fig. 17 is a second diagram showing a specific example of processing by the prediction and update step transformation unit. As shown in Fig. 17, the prediction and update step transformation unit 1524 linearizes the dynamics model, thereby transforming the prediction step 410 and generating a prediction step 1710.

[0101] The post-transformation prediction step 1710 is a step of estimating the distribution of the state of the virtual space after transformation at time t based on the observed values ​​from time 1 to time t-1. The post-transformation prediction step 1710 is described by a linear Gaussian distribution (dynamics model p(x ｔ |x ｔ－１) can be calculated analytically. Note that the update step 1720 cannot be calculated analytically because it is still described by a nonlinear Gaussian distribution.

[0102] (3) Specific Example of Processing by DBF Generator 1525 Next, a specific example of processing by the DBF generator 1525 of the information processing device 120 will be described. FIG. 18 is a second diagram showing a specific example of processing by the DBF generator. As shown in FIG. 18, the DBF generator 1525 introduces an inverse observation operator (reference numeral 1810), makes an update step including the inverse observation operator calculable, and generates a DBF (reference numeral 1820). Here, the content of the processing by the DBF generator 1525 will be described with reference to reference numeral 1800 (a diagram showing the relationship between each variable and parameter in a DBF).

[0103] The inverse observation operator r(x ｔ |o ｔ ) is an operator that estimates the distribution of the state of the virtual space after transformation at time t based on the observed value at time t (reference numeral 1800). As shown in reference numeral 1810, the inverse observation operator r(x ｔ |o ｔ ) includes the parameter of the operator f and the parameter of the operator G as parameters.

[0104] As shown in 1820, the DBF is the observations (o １：ｔ ), the state of the virtual space after transformation at time t (x ｔ ) calculated in the prediction step (observations (o) from time 1 to time t-1) １ : ｔ－１ ) and the state of the virtual space after transformation (x ｔ ) distribution) and the inverse observation operator r(x ｔ |o ｔ ) (observation value at time t (o ｔ ) and the state of the virtual space after transformation (x ｔ ) distribution) and the transformed virtual space state (x ｔ ) to calculate the posterior distribution.

[0105] (4) Specific Example of Processing by ELBO Generator 1526 Next, a specific example of processing by the ELBO generator 1526 of the information processing device 120 will be described. FIG. 19 is a second diagram showing a specific example of processing by the ELBO generator. As shown in FIG. 19, the ELBO generator 1526 generates an ELBO (reference numeral 1930), which is a function indicating the lower limit of the log-likelihood 1910 of the observation value. As shown by reference numeral 1930, the ELBO includes an approximation model q(x ｔ |o １：ｔ ) is included. The approximate model q(x ｔ |o １：ｔ ) is a DBF (reference numeral 1820). As described above, a DBF is a model that estimates a posterior distribution by recursively calculating a prediction step and an update step including an inverse observation operator.

[0106] The ELBO generated by the ELBO generating unit 1526 is input to the server device 110 and used to optimize the parameters included in the DBF.

[0107] As mentioned above, maximizing the ELBO minimizes the KL distance between the true posterior distribution and the DBF.

[0108] (5) Specific Example of Processing by Parameter Optimization Unit 112 Next, a specific example of processing by the parameter optimization unit 112 of the server device 110 will be described. Fig. 20 is a second diagram showing a specific example of processing by the parameter optimization unit. As shown in Fig. 20, the parameter optimization unit 112 has an ELBO maximization unit 1513, and optimizes parameters included in the DBF so as to maximize the ELBO generated by the ELBO generation unit 1526.

[0109] Specifically, the ELBO maximization unit 1513 maximizes the likelihood of the observation value by maximizing the inverse observation operator r(x ｔ |o ｔ ) parameters (parameters of operator f, parameters of operator G), linearized dynamics model p(x ｔ |x ｔ－１ ) parameters (parameter A, parameter Q) are optimized.

[0110] <Flow of Parameter Optimization Processing> Next, a description will be given of the flow of parameter optimization processing in the parameter optimization phase of the estimation system 1100 according to the second embodiment. Fig. 21 is an example of a second flowchart showing the flow of the parameter optimization processing.

[0111] In step S2101, the information processing apparatus 120 receives a state space model used to estimate a state in a physical space.

[0112] In step S2102, the information processing device 120 generates a prediction step including a dynamics model and an update step including an observation model.

[0113] In step S2103, the information processing device 120 linearizes the dynamics model described by the nonlinear Gaussian distribution using the Koopman operator, and describes the dynamics model by a linear Gaussian distribution.

[0114] In step S2104, the information processing device 120 describes the prediction step using a linear Gaussian distribution by introducing a dynamics model described using a linear Gaussian distribution into the prediction step.

[0115] In step S2105, the information processing apparatus 120 generates a DBF.

[0116] In step S2106, the information processing device 120 generates an ELBO for optimizing the parameters included in the DBF.

[0117] In step S2107, the server device 110 acquires the observation values ​​obtained by measuring the physical space.

[0118] In step S2108, the server device 110 inputs the observed values ​​and optimizes the parameters included in the DBF so as to maximize the ELBO.

[0119] <System Configuration of Estimation System (Estimation Phase)> Next, a system configuration in the estimation phase of the estimation system according to the second embodiment will be described. Fig. 22 is a diagram showing an example of the system configuration in the estimation phase of the estimation system according to the second embodiment. As shown in Fig. 22, an estimation system 2200 according to the second embodiment includes a measurement device 1140 and a server device 1110 (an example of an estimation device) in the estimation phase.

[0120] The measurement device 1140 is a device that measures the physical space 1130, and for example, the observed value o １ ~o ｔ Output.

[0121] An estimation program is installed in the server device 1110 , and when the program is executed, the server device 1110 functions as a posterior distribution estimation unit 2220 .

[0122] The posterior distribution estimation unit 2220 has a model in which parameters (parameters of the operator f, parameters of the operator G, parameter A, and parameter Q) are optimized. １ ~o ｔ Based on this, the state of the physical space at time t (z ｔ ) posterior distribution p(z ｔ |o １：ｔ ) is calculated.

[0123] <Specific Example of Processing by the Posterior Distribution Estimation Unit of the Estimation System (Estimation Phase)> Next, a specific example of processing by the posterior distribution estimation unit 2220 of the server device 1110 included in the estimation system 2200 in the inference phase will be described. FIG. 23 is a second diagram showing a specific example of processing by the posterior distribution estimation unit. As shown in FIG. 23 , the posterior distribution estimation unit 2220 estimates the observed values ​​(o １：ｔ ) at time t based on the transformed virtual space state (x ｔ ) posterior distribution p(x ｔ |o １：ｔ Specifically, the posterior distribution estimation unit 2220 estimates the observed value o１ ~o ｔ The model shown by the reference numeral 2221 is executed by inputting the following. The model shown by the reference numeral 2221 is an inverse observation operator r(x ｔ |o ｔ ) and the observed values ​​from time 1 to time t-1 (o １：ｔ－１ ) based on the transformed virtual space state (x ｔ ) distribution p(x ｔ |o １：ｔ－１ ) and the state of the virtual space after transformation at time t (x ｔ ) posterior distribution p(x ｔ |o １：ｔ ) is calculated.

[0124] Furthermore, the posterior distribution estimation unit 2220 calculates the operator (z ｔ =φ －１ (x ｔ )) and the formula for variable transformation of random variables, the state of the virtual space after transformation at time t (x ｔ ) posterior distribution p(x ｔ |o １：ｔ ) from the state at time t (z ｔ ) posterior distribution p(z ｔ |o １：ｔ ) is calculated.

[0125] <Flow of Estimation Process> Next, a description will be given of the flow of estimation process in the estimation phase of the estimation system 2200 according to the second embodiment. Fig. 24 is an example of a second flowchart showing the flow of the estimation process.

[0126] In step S2401, the server device 1110 acquires the observed values ​​measured by the measurement device 1140 from time 1 to time t.

[0127] In step S2402, the server device 1110 estimates the posterior distribution of the state of the physical space at time t based on the observed values ​​from time 1 to time t.

[0128] In step S2403, the server device 1110 outputs the posterior distribution of the state of the physical space based on the observed values ​​from time 1 to time t.

[0129] In step S2404, server device 1110 determines whether or not to continue the estimation process. If it is determined in step S2404 that the estimation process should be continued (YES in step S2404), the process returns to step S2401. In this case, server device 1110 advances time point t and executes the processes of steps S2401 to S2404. On the other hand, if it is determined in step S2404 that the estimation process should not be continued (NO in step S2404), the estimation process ends.

[0130] <Summary> As is clear from the above explanation, the estimation system 2200 according to the second embodiment calculates, in a state space model, a distribution p(z ｔ |z ｔ－１ ) is described by a linear Gaussian distribution using the Koopman operator, and the distribution p(o) of the observations at time t is determined based on the state at time t. ｔ |z ｔ ) is described by a nonlinear Gaussian distribution, when estimating the posterior distribution of the state at time t based on the observation values ​​up to time t, the posterior distribution of the state of physical space at time t based on the observation values ​​up to time t is calculated using a DBF that introduces an inverse observation operator and whose parameters are optimized to maximize the lower bound of the log-likelihood.

[0131] As described above, in the second embodiment, in a state space model, the prediction step is made calculable by linearizing the dynamics model using a Koopman operator, the update step is made calculable by introducing an inverse observation operator, and the posterior distribution is estimated by recursively calculating the prediction step and the update step to generate a DBF that estimates the posterior distribution. Also, in the second embodiment, parameters included in the DBF are optimized to maximize the lower bound of the log-likelihood. Then, in the second embodiment, the posterior distribution of the state is calculated using the DBF with optimized parameters.

[0132] As a result, according to the second embodiment, it is possible to reduce the computational cost required for performing estimation with the same accuracy as that of conventional posterior distribution estimation techniques. In other words, according to the second embodiment, it is possible to propose a new posterior distribution estimation technique that is more computationally cost effective than conventional posterior distribution estimation techniques.

[0133] [Third Embodiment] In the second embodiment, the prediction step is made calculable by linearizing the dynamics model using the Koopman operator. However, the method for making the prediction step calculable is not limited to this. For example, the dynamics model may be linearized by performing a Taylor expansion. The following describes the third embodiment, focusing on the differences from the first and second embodiments.

[0134] <System Configuration of Estimation System (Parameter Optimization Phase)> First, a system configuration in the parameter optimization phase of an estimation system according to the third embodiment will be described. Fig. 25 is a diagram showing an example of the system configuration in the parameter optimization phase of the estimation system according to the third embodiment. The differences between the system configuration in the parameter optimization phase of the estimation system 100 according to the second embodiment, which was described in the second embodiment using Fig. 15 , and the estimation system 2500 shown in Fig. 25 are as follows: the functions of the dynamics model linearizer 2523, the prediction and update step transformer 2524, the DBF generator 2525, and the ELBO generator 2526 included in the information processing device 120 are different; and the function of the ELBO maximizer 2513 of the parameter optimizer 112 included in the server device 110 is different.

[0135] The dynamics model linearization unit 2523 performs Taylor expansion on the dynamics model described by a nonlinear Gaussian distribution, thereby describing the dynamics model by a linear Gaussian distribution.

[0136] The prediction and update step modification unit 2524 describes the prediction step using a linear Gaussian distribution by introducing a dynamics model described using a linear Gaussian distribution into the prediction step of the prediction steps and update steps generated by the prediction and update step generation unit 122. This makes it possible to analytically calculate the prediction step. Note that at this stage, the update step cannot be analytically calculated because it is still described using a nonlinear Gaussian distribution.

[0137] The DBF generator 2525 generates an update step including an inverse observation operator, thereby making the update step calculable. Furthermore, the DBF generator 2525 generates a DBF that estimates a posterior distribution by recursively calculating a prediction step and an update step including an inverse observation operator.

[0138] The ELBO generator 2526 generates an ELBO in order to optimize the parameters included in the DBF. The ELBO generated by the ELBO generator 2526 is input to the server device 110 and used to optimize the parameters included in the DBF.

[0139] The parameter optimization unit 112 optimizes the parameters included in the DBF using the observed values ​​at each time point t stored in the observed value storage unit 1514. Specifically, the parameter optimization unit 112 has an ELBO maximization unit 2513, and optimizes the parameters included in the DBF so as to maximize the ELBO (lower bound of the log-likelihood) generated by the ELBO generation unit 2526.

[0140] <Specific Example of Processing by Each Unit of Estimation System (Parameter Optimization Phase)> Next, a specific example of processing by each unit of the server device 110 included in the estimation system 2500 in the parameter optimization phase and a specific example of processing by each unit of the information processing device 120 will be described. However, here, a specific example of processing by the ELBO maximization unit 2513 of the parameter optimization unit 112, among the units of the server device 110, will be described. Also, here, a specific example of processing by the dynamics model linearization unit 2523, the prediction and update step deformation unit 2524, the DBF generation unit 2525, and the ELBO generation unit 2526, among the units of the information processing device 120, will be described.

[0141] (1) Specific Example of Processing by the Dynamics Model Linearization Unit 2523 First, a specific example of processing by the dynamics model linearization unit 2523 of the information processing device 120 will be described. Fig. 26 is a third diagram showing a specific example of processing by the dynamics model linearization unit. The difference from the dynamics model linearization unit 1523 described with reference to Fig. 16 in the second embodiment is that when linearizing the dynamics model, the dynamics model is linearized by Taylor expansion. The dynamics model p(z ｔ |z ｔ－１ ) is written as a linear approximation formula and includes parameters A and Q (reference numeral 2610).

[0142] (2) Specific Example of Processing by the Prediction and Update Step Transformation Unit 2524 Next, a specific example of processing by the prediction and update step transformation unit 2524 of the information processing device 120 will be described. Fig. 27A is a third diagram showing a specific example of processing by the prediction and update step transformation unit. As shown in Fig. 27A, the prediction and update step transformation unit 2524 linearizes the dynamics model, thereby transforming the prediction step 410 and generating a prediction step 2701.

[0143] The prediction step 2701 after deformation is described by a linear Gaussian distribution (dynamics model p(z ｔ |z ｔ－１ ) can be calculated analytically. Note that the update step 420 after the transformation cannot be calculated analytically because it is still described by a nonlinear Gaussian distribution.

[0144] (3) Specific Example of Processing by DBF Generator 2525 Next, a specific example of processing by the DBF generator 2525 of the information processing device 120 will be described. FIG. 27B is a third diagram showing a specific example of processing by the DBF generator. As shown in FIG. 27B, the DBF generator 2525 introduces an inverse observation operator (reference numeral 2710), makes it possible to calculate an update step including the inverse observation operator, and generates a DBF (reference numeral 2720). Here, the contents of the processing by the DBF generator 2525 will be described with reference to reference numeral 2700 (a diagram showing the relationship between each variable and parameter in a DBF).

[0145] The inverse observation operator r(z ｔ |o ｔ ) is an operator that estimates the distribution of the state at time t based on the observed value at time t (reference numeral 2700). As shown in reference numeral 2710, the inverse observation operator r(z ｔ |o ｔ ) includes the parameter of the operator f and the parameter of the operator G as parameters.

[0146] As shown in 2720, the DBF is the observations (o １：ｔ ) based on the state (z ｔ) calculated in the prediction step (observations (o) from time 1 to time t-1) １ : ｔ－１ ) at time t estimated based on ｔ ) distribution) and the inverse observation operator r(z ｔ |o ｔ ) (observation value at time t (o ｔ ) at time t estimated based on ｔ ) and the state at time t based on ｔ ) to calculate the posterior distribution.

[0147] (4) Specific Example of Processing by the ELBO Generator 2526 Next, a specific example of processing by the ELBO generator 2526 of the information processing device 120 will be described. FIG. 27C is a third diagram showing a specific example of processing by the ELBO generator. As shown in FIG. 27B, the ELBO generator 2526 generates an ELBO (reference numeral 2730), which is a function indicating the lower limit of the log likelihood 1910 of the observation value. As shown by reference numeral 2730, the ELBO includes an approximation model q(z ｔ |o １：ｔ ) is included. The approximate model q(z ｔ |o １：ｔ ) is a DBF (reference numeral 2720). As described above, a DBF is a model that estimates a posterior distribution by recursively calculating a prediction step and an update step including an inverse observation operator.

[0148] The ELBO generated by the ELBO generating unit 2726 is input to the server device 110 and used to optimize the parameters included in the DBF.

[0149] As mentioned above, maximizing the ELBO minimizes the KL distance between the true posterior distribution and the DBF.

[0150] (5) Specific Example of Processing by Parameter Optimization Unit 112 Next, a specific example of processing by the parameter optimization unit 112 of the server device 110 will be described. Fig. 28 is a third diagram showing a specific example of processing by the parameter optimization unit. As shown in Fig. 28, the parameter optimization unit 112 has an ELBO maximization unit 2513, and optimizes parameters included in the DBF so as to maximize the ELBO generated by the ELBO generation unit 2526.

[0151] Specifically, the ELBO maximization unit 2513 maximizes the likelihood of the observation value by maximizing the inverse observation operator r(z ｔ |o ｔ ) parameters (parameters of operator f, parameters of operator G), linearized dynamics model p(z ｔ |z ｔ－１ ) parameters (parameter A, parameter Q) are optimized instead.

[0152] <Flow of Parameter Optimization Processing> Next, a description will be given of the flow of parameter optimization processing in the parameter optimization phase of the estimation system 2500 according to the third embodiment. Fig. 29 is an example of a third flowchart showing the flow of the parameter optimization processing.

[0153] In step S2901, the information processing apparatus 120 receives a state space model used to estimate a state in a physical space.

[0154] In step S2902, the information processing device 120 generates a prediction step including a dynamics model and an update step including an observation model.

[0155] In step S2903, the information processing device 120 linearizes the dynamics model described by the nonlinear Gaussian distribution by Taylor expansion, and describes the dynamics model by a linear Gaussian distribution.

[0156] In step S2904, the information processing device 120 describes the prediction step using a linear Gaussian distribution by introducing a dynamics model described using a linear Gaussian distribution into the prediction step.

[0157] In step S2905, the information processing apparatus 120 generates a DBF.

[0158] In step S2906, the information processing device 120 generates an ELBO for optimizing the parameters included in the DBF.

[0159] In step S2907, the server device 110 acquires the observation values ​​obtained by measuring the physical space.

[0160] In step S2908, the server device 110 inputs the observed values ​​and optimizes the parameters included in the DBF so as to maximize the ELBO.

[0161] <System Configuration of Estimation System (Estimation Phase)> The system configuration of the estimation system according to the third embodiment in the estimation phase is similar to the system configuration of the estimation system 2200 according to the second embodiment, and therefore description thereof will be omitted here.

[0162] <Summary> As is clear from the above explanation, the estimation system 2200 according to the third embodiment calculates, in a state space model, a distribution p(z ｔ |z ｔ－１ ) is described by a linear Gaussian distribution through Taylor expansion, and the distribution of observations at time t, p(o ｔ |z ｔ ) is described by a nonlinear Gaussian distribution, when estimating the posterior distribution of the state at time t based on the observed values ​​up to time t, the posterior distribution of the state at time t based on the observed values ​​up to time t is calculated using a DBF that introduces an inverse observation operator and whose parameters are optimized to maximize the lower bound of the log-likelihood.

[0163] As described above, in the third embodiment, in a state space model, the prediction step is made calculable by linearizing the dynamics model using a Taylor expansion, the update step is made calculable by introducing an inverse observation operator, and the posterior distribution is estimated by recursively calculating the prediction step and the update step to generate a DBF that estimates the posterior distribution. Furthermore, in the third embodiment, parameters included in the DBF are optimized to maximize the lower bound of the log-likelihood. Then, in the third embodiment, the posterior distribution of the state is calculated using the DBF with optimized parameters.

[0164] As a result, according to the third embodiment, it is possible to reduce the computational cost required for performing estimation with the same accuracy as that of conventional posterior distribution estimation techniques. In other words, according to the third embodiment, it is possible to propose a new posterior distribution estimation technique that is more computationally cost effective than conventional posterior distribution estimation techniques.

[0165] In the second and third embodiments, the dynamics model is described as a nonlinear Gaussian distribution in which the operator is nonlinear and the process noise follows a Gaussian distribution. In contrast, in the fourth embodiment, the dynamics model is described as a linear Gaussian distribution in which the operator is linear and the process noise follows a Gaussian distribution.

[0166] <System Configuration of Estimation System (Parameter Optimization Phase)> First, a system configuration in the parameter optimization phase of an estimation system according to the fourth embodiment will be described. Fig. 30 is a diagram showing an example of a system configuration in the parameter optimization phase of an estimation system according to the fourth embodiment. The differences between the system configuration in the parameter optimization phase of the estimation system 2500 according to the third embodiment described above using Fig. 25 in the third embodiment and the estimation system 3000 shown in Fig. 30 are as follows: The functions of a state space model input unit 3021 and a prediction and update step generation unit 3022 included in the information processing device 120 are different.

[0167] The state space model input unit 3021 inputs a state space model used to estimate a state in the physical space 1130. In the fourth embodiment, the dynamics model constituting the state space model is described by a linear Gaussian distribution, in which the operator is linear and the observation noise follows a Gaussian distribution. Also in the fourth embodiment, the observation model constituting the state space model is described by a nonlinear Gaussian distribution, in which the operator is nonlinear and the observation noise follows a Gaussian distribution. Note that in the fourth embodiment, it is assumed that the parameters of the dynamics model and the parameters of the observation model are unknown in the state space model.

[0168] The prediction and update step generator 3022 generates a prediction step including a dynamics model and an update step including an observation model. The prediction step generated by the prediction and update step generator 3022 is described by a linear Gaussian distribution, and the update step is described by a nonlinear Gaussian distribution.

[0169] <Specific Example of Processing by Each Unit of Estimation System (Parameter Optimization Phase)> Next, a specific example of processing by each unit of the server device 110 included in the estimation system 3000 in the parameter optimization phase and a specific example of processing by each unit of the information processing device 120 will be described. However, here, a specific example of processing by the prediction and update step generation unit 3022 of the information processing device 120 will be described.

[0170] 31 is a diagram showing a specific example of processing by the prediction and update step generator 3022. As shown in FIG. 31, the prediction and update step generator 3022 generates prediction steps 3110 and update steps 420.

[0171] The prediction step 3110 predicts the observed values ​​(o １：ｔ－１ ) based on the state (z ｔ ) distribution estimation step p(z ｔ |o １：ｔ－１ Specifically, the prediction step 3110 is performed by: - ​​calculating the results calculated in the update step (observed values ​​(o) from time 1 to time t-1). １：ｔ－１ ) at time t estimated based on ｔ－１) posterior distribution p(z ｔ－１ |o １：ｔ－１ )) and the result calculated by the dynamics model (state (z ｔ－１ ) at time t estimated based on ｔ )) and . The prediction step 3110 can be calculated analytically because it is described by a linear Gaussian distribution. Note that the subsequent processing in the parameter optimization phase (processing by the DBF generation unit 2525, ELBO generation unit 2526, and parameter optimization unit 112) is the same as in the third embodiment, and therefore description thereof will be omitted here.

[0172] <System configuration of estimation system (estimation phase)> The system configuration of the estimation system according to the fourth embodiment in the estimation phase is similar to the system configuration of the estimation system 2200 according to the second and third embodiments, and therefore will not be described here.

[0173] <Summary> As is clear from the above explanation, the estimation system 2200 according to the fourth embodiment calculates, in a state space model, a distribution p(z ｔ |z ｔ－１ ) is described by a linear Gaussian distribution, and the distribution of observations at time t, p(o ｔ |z ｔ ) is described by a nonlinear Gaussian distribution, when estimating the posterior distribution of the state at time t based on the observed values ​​up to time t, the posterior distribution of the state at time t based on the observed values ​​up to time t is calculated using a DBF that introduces an inverse observation operator and whose parameters are optimized to maximize the lower bound of the log-likelihood.

[0174] As described above, in the fourth embodiment, in a state space model, the prediction step is made calculable by describing the dynamics model with a linear Gaussian distribution, the update step is made calculable by introducing an inverse observation operator, and the posterior distribution is estimated by generating a DBF that estimates the posterior distribution by recursively calculating the prediction step and the update step. Also, in the fourth embodiment, the parameters included in the approximation model of the DBF are optimized to maximize the lower bound of the log-likelihood. Then, in the fourth embodiment, the posterior distribution of the state is calculated using the DBF with optimized parameters.

[0175] As a result, according to the fourth embodiment, it is possible to reduce the computational cost required for performing estimation with the same accuracy as that of conventional posterior distribution estimation techniques. In other words, according to the fourth embodiment, it is possible to propose a new posterior distribution estimation technique that is more computationally cost effective than conventional posterior distribution estimation techniques.

[0176] Fifth Embodiment In the above-described first to fourth embodiments, a case has been described in which a prediction step and an update step are used to estimate a posterior distribution of a state at time t based on observed values ​​up to time t. In contrast, in the fifth embodiment, a case will be described in which a so-called smoothing distribution is used to estimate a posterior distribution of a state at time t.

[0177] However, in the fifth embodiment, when estimating the posterior distribution of the state at time t using a smoothing distribution, an approximation model is generated using temporally or spatially local observation values. Also, in the fifth embodiment, when optimizing parameters included in the approximation model, the calculation is made possible by assuming predetermined conditions, and a parameter-optimized smoothing distribution is generated.

[0178] Furthermore, in the fifth embodiment, a case will be described in which a posterior distribution of a state at time t is estimated by using a filtered distribution generated based on a parameter-optimized smoothing distribution. However, in the fifth embodiment, when estimating the posterior distribution of a state at time t using the filtered distribution, an approximation model is generated using temporally or spatially local observation values, similar to the smoothing distribution. Also, in the fifth embodiment, when optimizing the parameters included in the approximation model, a relational expression between the smoothing distribution and the filtered distribution is used. Furthermore, when optimizing the parameters included in the generated approximation model, similar to the smoothing distribution, the parameters are made calculable by assuming predetermined conditions, and a parameter-optimized smoothing distribution is generated.

[0179] The fifth embodiment will be described in detail below, focusing on the differences from the first to fourth embodiments.

[0180] <System Configuration of Estimation System (Parameter Optimization Phase)> First, a system configuration in the parameter optimization phase of an estimation system according to the fifth embodiment will be described. Fig. 32 is a diagram showing an example of a system configuration in the parameter optimization phase of an estimation system according to the fifth embodiment. The differences between the system configuration in the parameter optimization phase of the estimation system 2500 according to the third embodiment, which was described in the third embodiment using Fig. 25 , and the estimation system 3200 shown in Fig. 32 are that: the information processing device 120 has a smoothing distribution generation unit 3221, an ELBO generation unit 3222, a calculability condition input unit 3223, a filtering distribution generation unit 3224, a KL distance generation unit 3225, and a calculability condition input unit 3226; the function of the ELBO maximization unit 3213 of the parameter optimization unit 112 of the server device 110 is different; and the parameter optimization unit 112 of the server device 110 has a KL distance minimization unit 3214.

[0181] The smoothing distribution generator 3221 generates a smoothing distribution that estimates the posterior distribution of the state at time t, based on 2N+1 observation values ​​(local observation values ​​in the time direction relative to time t) from time t-N to time t+N, which are time points before and after time t. Note that, although a case where a smoothing distribution is generated based on local observation values ​​in the time direction relative to time t will be described here, a smoothing distribution may also be generated based on local observation values ​​in the space direction relative to a predetermined measurement point at time t.

[0182] The ELBO generation unit 3222 generates an ELBO used when optimizing parameters included in the smoothing distribution. The ELBO generated by the ELBO generation unit 3222 is transmitted to the server device 110 and used to optimize parameters included in the smoothing distribution.

[0183] The calculability condition input unit 3223 inputs conditions for enabling calculations to optimize parameters included in the smoothing distribution using the ELBO generated by the ELBO generation unit 3222. When optimizing parameters included in a smoothing distribution based on observation values ​​local in the time direction for time point t, the calculability condition input unit 3223 inputs conditions specifying the relationship between states in the time direction. When optimizing parameters included in a smoothing distribution based on observation values ​​local in the space direction for a predetermined measurement point at time point t, the calculability condition input unit 3223 inputs conditions specifying the relationship between states in the space direction. The input calculability conditions are transmitted to the server device 110, and the ELBO maximization unit 3213 optimizes parameters included in the smoothing distribution under the assumption that the states in the time direction or the space direction have the specified relationship.

[0184] The filtering distribution generation unit 3224 generates a filtering distribution that estimates the posterior distribution of the state at time t, based on N+1 observation values ​​from time t-N to time t (observation values ​​that are local in the time direction with respect to time t). Note that, although a case where a filtering distribution is generated based on observation values ​​that are local in the time direction with respect to time t will be described here, a filtering distribution may also be generated based on observation values ​​that are local in the space direction with respect to a predetermined measurement point at time t.

[0185] The KL distance generation unit 3225 generates a KL distance, which is a measure for optimizing parameters included in the filtering distribution. The KL distance generation unit 3225 generates the KL distance using a relational expression between the smoothing distribution and the filtering distribution. Therefore, the KL distance includes the adjusted smoothing distribution. The KL distance generated by the KL distance generation unit 3225 is transmitted to the server device 110 and used for optimizing parameters included in the filtering distribution.

[0186] The computability condition input unit 3226 inputs conditions for enabling calculations to optimize parameters included in the filtering distribution using the KL distance generated by the KL distance generation unit 3225. When optimizing parameters included in the filtering distribution based on observation values ​​local in the time direction with respect to time t, the computability condition input unit 3226 inputs conditions specifying the relationship between states in the time direction. Furthermore, when optimizing parameters included in the filtering distribution based on observation values ​​local in the space direction with respect to a predetermined measurement point at time t, the computability condition input unit 3226 inputs conditions specifying the relationship between states in the space direction. The input computability conditions are transmitted to the server device 110, and the KL distance minimization unit 3214 optimizes the parameters included in the filtering distribution under the assumption that the states in the time direction or the space direction have the specified relationship.

[0187] The parameter optimization unit 112 has an ELBO maximization unit 3213 and a KL distance minimization unit 3214, and optimizes the parameters included in the smoothing distribution and the filtering distribution using the observed values ​​at each time point t stored in the observed value storage unit 2514.

[0188] Specifically, the ELBO maximization unit 3213 optimizes the parameters included in the smoothing distribution so as to maximize the ELBO (lower bound of the log-likelihood) transmitted by the ELBO generation unit 3222 under the calculability conditions transmitted by the calculability condition input unit 3223. The ELBO maximization unit 3213 transmits the parameter-optimized smoothing distribution, in which the parameters have been optimized, to the information processing device 120.

[0189] The KL distance minimization unit 3214 optimizes the parameters included in the filtering distribution so as to minimize the KL distance transmitted by the KL distance generation unit 3225 under the computation enabling conditions transmitted by the computation enabling condition input unit 3226.

[0190] <Specific Example of Processing by Each Unit of Estimation System (Parameter Optimization Phase)> Next, a specific example of processing by each unit of the server device 110 included in the estimation system 3200 in the parameter optimization phase and a specific example of processing by each unit of the information processing device 120 will be described. However, here, a specific example of processing by the smoothing distribution generation unit 3221 to the calculation enabling condition input unit 3226 among the units of the information processing device 120 will be described. Also, a specific example of processing by the ELBO maximization unit 3213 and the KL distance minimization unit 3214 among the units of the server device 110 will be described.

[0191] (1) Specific Example of Processing by Smoothing Distribution Generator 3221 First, a specific example of processing by the smoothing distribution generator 3221 of the information processing device 120 will be described. Fig. 33 is a diagram showing a specific example of processing by the smoothing distribution generator. The smoothing distribution generator 3221 generates a smoothing distribution 3310.

[0192] 33 , a typical smoothing distribution estimates the posterior distribution of the state at time t based on observed values ​​from time 1 to time T (T>t). In contrast, the smoothing distribution 3310 generated by the smoothing distribution generation unit 3221 calculates the posterior distribution of the state at time t based on observed values ​​localized in the time direction relative to time t, using a function that is invariant regardless of time.

[0193] Specifically, the smoothing distribution 3310 generated by the smoothing distribution generation unit 3221 is an approximation model (first model) that calculates the posterior distribution of the state at time t based on 2N+1 observed values ​​from time t−N to time t+N, which are the time points before and after time t. μ ｓ and Σ ｓ is an operator that estimates the posterior distribution of the state at time t based on the observed values ​​from time t−N to time t+N, and is invariant regardless of time.

[0194] In addition, the smoothing distribution generation unit 3221 may generate a smoothing distribution (not shown) that calculates the posterior distribution of the state at time t based on local observation values ​​in the spatial direction for a specified measurement point at time t using a function that is invariant regardless of spatial position.

[0195] (2) Specific Example of Processing by ELBO Generator 3222 Next, a specific example of processing by the ELBO generator 3222 of the information processing device 120 will be described. Figures 34A and 34B are first and second diagrams showing a specific example of processing by the ELBO generator. The ELBO generator 3222 generates an ELBO 3410 using the smoothing distribution 3310 generated by the smoothing distribution generator 3221.

[0196] As shown in Fig. 34A, ELBO 3410 requires input of estimated values ​​calculated using a filtering distribution (see the dotted rectangle in Fig. 34A). However, at the stage of optimizing the parameters of smoothing distribution 3310 using ELBO 3410, a parameter-optimized filtering distribution has not yet been generated, making it difficult to input highly accurate estimated values. Therefore, instead of inputting estimated values ​​calculated using a filtering distribution, constants based on each time series of various observed values ​​(e.g., the mean value and variance of the state at time 0) may be input.

[0197] Also, as shown in FIG. 34B, the fourth and sixth terms of ELBO 3410, q(z ｔ |o －Ｎ：Ｔ ) is calculated using the approximation shown at 3420.

[0198] (3) Specific Example of Processing by the Computation Enabling Condition Input Unit 3223 Next, a specific example of processing by the computation enabling condition input unit 3223 of the information processing device 120 will be described. Fig. 35 is a first diagram showing a specific example of processing by the computation enabling condition input unit. The computation enabling condition input unit 3223 inputs computation enabling conditions 3510 for enabling calculations that optimize parameters included in the smoothing distribution 3310, using the ELBO 3410 generated by the ELBO generation unit 3222.

[0199] The calculability condition 3510 is a relational expression that indicates the assumption that the state of the jth component at time t+k has a correlation that is calculated by multiplying the state of the i-th component at time t by a constant attenuation coefficient for the time interval k, regardless of the time. Specifically, the calculability condition 3510 is: The state of the i-th component at time t (z ｉ＿ｔ ) and the state of the i-th component at time t (z ｉ ) and the difference value between the average value of the j-th component at time t+k (z ｊ＿ｔ＋ｋ ) and the state of the jth component at time t+k (z ｊ ) and the product of the standard deviation of the distribution of the state of the i-th component at time t, and the standard deviation of the distribution of the state of the j-th component at time t+k, is equal to the product of the attenuation coefficient ρ raised to the power of |k| (where ρ is a value between 0 and 1). Note that the attenuation coefficient ρ is "0" when the state of the i-th component at time t and the state of the j-th component at time t+k are independent of each other.

[0200] By assuming the correlation shown in the calculation enabling condition 3510, the variance-covariance matrix is ​​expressed as the variance-covariance matrix 3520 in FIG. 35, making it possible to calculate the ELBO 3410.

[0201] (4) Specific Example of Processing by ELBO Maximizer 3213 Next, a specific example of processing by the ELBO maximizer 3213 of the parameter optimizer 112 of the server device 110 will be described. Fig. 36 is a fourth diagram showing a specific example of processing by the parameter optimizer. As shown in Fig. 36, the ELBO maximizer 3213 of the parameter optimizer 112 optimizes parameters included in the smoothing distribution 3310 using ELBO 3410 under a calculation enabling condition 3510.

[0202] Specifically, the ELBO maximization unit 3213 maximizes the observed value (o ｔ－Ｎ ~o ｔ＋Ｎ ) contained in the smoothing distribution 3310 so as to maximize the likelihood of ｓ parameters and Σ ｓ Optimize the parameters of

[0203] (5) Specific Example of Processing by Filtering Distribution Generation Unit 3224 Next, a specific example of processing by the filtering distribution generation unit 3224 of the information processing device 120 will be described. Fig. 37 is a diagram showing a specific example of processing by the filtering distribution generation unit. The filtering distribution generation unit 3224 generates a filtering distribution 3710.

[0204] 37 , a typical filtering distribution estimates the posterior distribution of the state at time t based on observed values ​​from time 1 to time t. In contrast, the filtering distribution 3710 generated by the filtering distribution generation unit 3224 calculates the posterior distribution of the state at time t based on observed values ​​local in the time direction relative to time t, using a function that is invariant regardless of time.

[0205] Specifically, the filtering distribution 3710 generated by the filtering distribution generation unit 3224 is an approximation model (second model) that estimates the posterior distribution of the state at time t based on N+1 observed values ​​from time t−N to time t, which are the time points up to time t. μ ｆ and Σ ｆ is an operator that estimates the posterior distribution of the state at time t based on the observed values ​​from time t−N to time t, and is invariant regardless of time.

[0206] In addition, the filtering distribution generation unit 3224 may generate a filtering distribution (not shown) that calculates the posterior distribution of the state at time t based on local observation values ​​in the spatial direction for a specified measurement point at time t using a function that is invariant regardless of spatial position.

[0207] (6) Specific Example of Processing by KL Distance Generation Unit 3225 Next, a specific example of processing by the KL distance generation unit 3225 of the information processing device 120 will be described. Fig. 38 is a diagram showing a specific example of processing by the KL distance generation unit. The KL distance generation unit 3225 generates a KL distance 3810, which is a measure for optimizing parameters included in the filtering distribution. The KL distance generation unit 3225 generates the KL distance 3810 by using a relational expression between the smoothing distribution and the filtering distribution.

[0208] The KL distance 3810 is calculated using: - the posterior distribution of the state at time t estimated based on the observations from time t-N to time t+N (i.e., the calculation result using the adjusted smoothing distribution); - the distribution of observations from time t+1 to time t+N estimated based on the observations from time t-N to time t; - the posterior distribution at time t estimated based on the observations from time t-N to time t (i.e., the calculation result using the filtered distribution).

[0209] (7) Specific Example of Processing by the Computational Condition Input Unit 3226 Next, a specific example of processing by the computational condition input unit 3226 of the information processing device 120 will be described. Fig. 39 is a second diagram showing a specific example of processing by the computational condition input unit. The computational condition input unit 3226 inputs a computational condition 3910 for enabling a calculation to optimize parameters included in the filtering distribution 3710, using the KL distance 3810 generated by the KL distance generation unit 3225.

[0210] The calculability condition 3910 is a relational expression that indicates the assumption that the state of the jth component at time t+k has a correlation that is calculated by multiplying the state of the i-th component at time t by a constant attenuation coefficient for the time interval k, regardless of the time. Specifically, the calculability condition 3910 is: The state of the i-th component at time t (z ｉ＿ｔ ) and the state of the i-th component at time t (z ｉ ) and the difference value between the average value of the j-th component at time t+k (z ｊ＿ｔ＋ｋ ) and the state of the jth component at time t−k (z ｊ ) and the integrated value of ｔ ) and the standard deviation of the state of the jth component at time t + k (σ ｔ－ｋ ) and the value obtained by multiplying the value obtained by raising the attenuation coefficient ρ to the power |k| (where ρ is a value between 0 and 1). Note that the attenuation coefficient ρ is "0" when the state of the i-th component at time t and the state of the j-th component at time t+k are independent of each other.

[0211] By assuming the correlation shown in the calculation enabling condition 3910, the variance-covariance matrix is ​​expressed as the variance-covariance matrix 3920 in FIG. 38, making it possible to calculate the KL distance 3810.

[0212] (8) Specific Example of Processing by KL Distance Minimization Unit 3214 Next, a specific example of processing by the KL distance minimization unit 3214 of the parameter optimization unit 112 of the server device 110 will be described. Fig. 40 is a fifth diagram showing a specific example of processing by the parameter optimization unit. As shown in Fig. 40 , the KL distance minimization unit 3214 of the parameter optimization unit 112 optimizes parameters included in the filtering distribution 3710 under a calculability condition 3910 so as to minimize the KL distance 3810.

[0213] Specifically, the KL distance minimization unit 3214 calculates the observed value (o ｔ－Ｎ ~o ｔ) is input. The KL distance minimization unit 3214 inputs the sum of the KL distances 3810 from t=0 to t=T−N. ｆ parameters and Σ ｆ Optimize the parameters of

[0214] <Flow of Parameter Optimization Processing> Next, a description will be given of the flow of parameter optimization processing in the parameter optimization phase of the estimation system 3200 according to the fifth embodiment. Fig. 41 is an example of a fourth flowchart showing the flow of the parameter optimization processing.

[0215] In step S4101, the information processing device 120 receives a state space model used to estimate a state in a physical space.

[0216] In step S4102, the information processing device 120 generates a smoothing distribution. Specifically, the information processing device 120 generates an approximation model (first approximation model) that calculates the posterior distribution of the state at time t based on observed values ​​in a local time range before and after time t, using a function that is invariant regardless of time. Alternatively, the information processing device 120 generates an approximation model (first approximation model) that calculates the posterior distribution of the state at time t based on observed values ​​in a local spatial range around a predetermined measurement point at time t, using a function that is invariant regardless of spatial position.

[0217] In step S4103, the information processing device 120 generates an ELBO for optimizing the parameters included in the generated smoothing distribution.

[0218] In step S4104, the information processing device 120 inputs a calculation enabling condition that enables calculation for optimizing the parameters included in the smoothing distribution using ELBO.

[0219] In step S4105, the server device 110 acquires the observation values ​​obtained by measuring the physical space.

[0220] In step S4106, the server device 110 inputs the observed values, updates the parameters included in the smoothing distribution so as to maximize the ELBO under the calculation-enabling conditions, and generates a parameter-optimized smoothing distribution.

[0221] In step S4107, the information processing device 120 generates a filtering distribution. Specifically, the information processing device 120 generates an approximation model (second approximation model) that calculates the posterior distribution of the state at time t based on observed values ​​in a local time range up to time t, using a function that is invariant regardless of time. Alternatively, the information processing device 120 generates an approximation model (second approximation model) that calculates the posterior distribution of the state at time t based on observed values ​​in a local spatial range around a predetermined measurement point at time t, using a function that is invariant regardless of spatial position.

[0222] In step S4108, the information processing device 120 generates a KL distance for optimizing the parameters included in the generated filtering distribution.

[0223] In step S4109, the information processing device 120 inputs a calculation enabling condition that enables calculation for optimizing parameters included in the filtering distribution using the KL distance.

[0224] In step S4110, the server device 110 acquires the observed values ​​obtained by measuring the physical space.

[0225] In step S4111, the server device 110 inputs the observed values, and optimizes the parameters included in the filtering distribution so as to minimize the KL distance under the calculation enabling conditions, thereby generating a parameter-optimized filtering distribution.

[0226] <System Configuration of Estimation System (Estimation Phase)> Next, a system configuration in the estimation phase of an estimation system according to the fifth embodiment will be described. Fig. 42 is a diagram showing an example of the system configuration in the estimation phase of the estimation system according to the fifth embodiment. As shown in Fig. 42, an estimation system 4200 according to the fifth embodiment includes a measurement device 1140 and a server device 1110 (an example of an estimation device) in the estimation phase.

[0227] The measurement device 1140 is a device that measures the physical space 1130, and for example, the observed value o ｔ－Ｎ ~o ｔ＋Ｎ Alternatively, the measurement device 1140 outputs the observed value o ｔ－Ｎ ~o ｔ Output.

[0228] An estimation program is installed in the server device 1110, and by executing this program, the server device 1110 functions as a posterior distribution estimation unit 4220 and a posterior distribution estimation unit 4230.

[0229] The posterior distribution estimator 4220 has a parameter-optimized smoothing distribution. The posterior distribution estimator 4220 estimates the observed value o measured by the measurement device 1140. ｔ－Ｎ ~o ｔ＋Ｎ Based on this, the posterior distribution q(z ｔ |o ｔ－Ｎ：ｔ＋Ｎ ) is estimated.

[0230] The posterior distribution estimator 4230 has a parameter-optimized filtering distribution. ｔ－Ｎ ~o ｔ Based on this, the posterior distribution q(z ｔ |o ｔ－Ｎ：ｔ ) is estimated.

[0231] <Specific Example of Processing by Posterior Distribution Estimation Unit of Estimation System (Estimation Phase)> Next, a specific example of processing by the posterior distribution estimation unit 4220 and the posterior distribution estimation unit 4230 of the server device 1110 provided in the estimation system 4200 in the inference phase will be described. Fig. 43 is a third diagram showing a specific example of processing by the posterior distribution estimation unit. As shown in Fig. 43, the posterior distribution estimation unit 4220 calculates μ ｓ and Σ ｓ The parameters of the model q(z ｔ |o ｔ－Ｎ：ｔ＋Ｎ ) model q(z ｔ |o ｔ－Ｎ：ｔ＋Ｎ ) is the observed value (o ｔ－Ｎ：ｔ＋Ｎ ) based on the state (z ｔ ) to estimate the posterior distribution.

[0232] The posterior distribution estimation unit 4230 calculates μ ｆ and Σ ｆ The parameters of the model q(z ｔ |o ｔ－Ｎ：ｔ ) model q(z ｔ |o ｔ－Ｎ：ｔ ) is the observed value (o ｔ－Ｎ：ｔ ) based on the state (z ｔ ) to estimate the posterior distribution.

[0233] <Flow of Estimation Processing> Next, the flow of estimation processing in the estimation phase of the estimation system 4200 according to the fifth embodiment will be described. Fig. 44 is an example of a third flowchart showing the flow of the estimation processing. Of these, Fig. 44(a) is a flowchart showing the flow of the estimation processing by the posterior distribution estimation unit 4220, and Fig. 44(b) is a flowchart showing the flow of the estimation processing by the posterior distribution estimation unit 4230.

[0234] In step S4401, the posterior distribution estimation unit 4220 of the server device 1110 acquires the observed values ​​measured by the measurement device 1140 from time t−N to time t+N.

[0235] In step S4402, the posterior distribution estimation unit 4220 of the server device 1110 estimates the posterior distribution of the state of the physical space at time t based on the observed values ​​from time t−N to time t+N.

[0236] In step S4403, the server device 1110 outputs the posterior distribution of the state of the physical space at time t based on the observed values ​​from time t−N to time t+N.

[0237] In step S4404, server device 1110 determines whether or not to continue the estimation process. If it is determined in step S4404 that the estimation process should be continued (YES in step S4404), the process returns to step S4401. In this case, server device 1110 advances time t and then executes the processes of steps S4401 to S4404. On the other hand, if it is determined in step S4404 that the estimation process should not be continued (NO in step S4404), the estimation process ends.

[0238] In step S4411, the posterior distribution estimation unit 4230 of the server device 1110 acquires the observed values ​​measured by the measurement device 1140 from time t−N to time t.

[0239] In step S4412, the posterior distribution estimation unit 4230 of the server device 1110 estimates the posterior distribution of the state of the physical space at time t based on the observed values ​​from time t−N to time t.

[0240] In step S4413, the server device 1110 outputs the posterior distribution of the state of the physical space at time t based on the observed values ​​from time t−N to time t.

[0241] In step S4414, server device 1110 determines whether or not to continue the estimation process. If it is determined in step S4414 that the estimation process should be continued (YES in step S4414), the process returns to step S4411. In this case, server device 1110 advances time point t and then executes the processes of steps S4411 to S4414. On the other hand, if it is determined in step S4414 that the estimation process should not be continued (NO in step S4414), the estimation process ends.

[0242] <Summary> As is clear from the above explanation, when the estimation system 4200 according to the fifth embodiment uses a state space model to estimate the posterior distribution of the state at time t based on observation values ​​up to time t, the posterior distribution of the state at a given time is calculated using: an approximation model that calculates the posterior distribution of the state at a given time based on observation values ​​that are local in the time direction for the given time point or observation values ​​that are local in the space direction for a given measurement point at the given time point, using a function that is invariant regardless of time and spatial position, and an approximation model whose parameters are optimized under the assumption that the distribution of the state at the given time point has a correlation that is calculated by multiplying a constant attenuation coefficient for a time interval regardless of time, or an approximation model whose parameters are optimized under the assumption that the distribution of the state at a given measurement point at a given time point has a correlation that is calculated by multiplying a constant attenuation coefficient for a spatial interval regardless of spatial position.

[0243] As described above, in the fifth embodiment, a parameter-optimized smoothing distribution (first approximation model) or a parameter-optimized filtered distribution (second approximation model) is generated to estimate the posterior distribution of the state at time t using local observation values ​​in the time direction or the spatial direction. In this case, in the fifth embodiment, the parameters included in the smoothing distribution and the parameters included in the filtered distribution are optimized by assuming predetermined conditions and making them computable. Then, in the fifth embodiment, the parameter-optimized first approximation model or second approximation model is used to calculate the posterior distribution of the state at time t.

[0244] As a result, according to the fifth embodiment, it is possible to reduce the computational cost required for performing estimation with the same accuracy as that of conventional posterior distribution estimation techniques. In other words, according to the fifth embodiment, it is possible to propose a new posterior distribution estimation technique that is more computationally cost effective than conventional posterior distribution estimation techniques.

[0245] Sixth Embodiment In the first to fifth embodiments, variations of the new posterior distribution estimation technique have been described. In contrast, in the sixth embodiment, an application example to which the new posterior distribution estimation technique is applied will be described. Here, as an application example to which the posterior distribution estimation technique described in the fifth embodiment is applied, an application example to the meteorological field (specifically, estimation of the posterior distribution of parameters of raindrop particle size distribution in a precipitation space) will be described. Note that the present disclosure may also be applied to weather forecasting, such as estimating the amount of water vapor or ice contained in the atmosphere, or the atmospheric temperature or pressure.

[0246] <System Configuration of Raindrop Particle Size Distribution Estimation System (Parameter Optimization Phase)> Fig. 45 is a diagram showing an example of the system configuration of the raindrop particle size distribution estimation system in the parameter optimization phase. As shown in Fig. 45, the raindrop particle size distribution estimation system 4500 in the parameter optimization phase includes a dual-polarization radar 4540, a server device 110, and an information processing device 120.

[0247] The dual-polarized radar 4540 measures the rain volume 4530 and obtains observations useful for estimating parameters of the drop size distribution (DSD), a physical variable related to the amount of precipitation. ｔ As the horizontally polarized radar reflectivity factor Z Ｈ (z ｔ ), and the difference in reflectivity factor between polarizations Z ｄｒ (z ｔ ), ・Phase difference between polarizations Φ ＤＰｔ , and the observation value o output by the dual polarization radar 4540 is ｔ is stored in the observation value storage unit 2514 of the server device 110.

[0248] The state space model input unit 121 inputs a state space model used to estimate the state in the rainfall space 4530. In the sixth embodiment, the state space model is a dynamics model p(z ｔ＋１ |z ｔ ) and the observation model p(o ｔ |z ｔ )

[0249] The smoothing distribution generator 3221 generates a smoothing distribution. As described in the fifth embodiment, the smoothing distribution generated by the smoothing distribution generator 3221 is an approximation model using temporally local observation values, and estimates parameters of the raindrop particle diameter distribution in the rainfall space.

[0250] The observed value o measured in the raindrop particle size distribution estimation system 4500 ｔ are observed values ​​measured at, for example, 800 measurement points, and the number of observed values ​​is finite. Therefore, the parameter optimization unit 112 of the raindrop particle diameter distribution estimation system 4500 optimizes the parameters included in the smoothing distribution without using ELBO.

[0251] <System Configuration of Raindrop Particle Diameter Distribution Estimation System (Estimation Phase)> Fig. 46 is a diagram showing an example of a system configuration in the estimation phase of the raindrop particle diameter distribution estimation system. As shown in Fig. 46, in the estimation phase, the raindrop particle diameter distribution estimation system 4600 includes a dual-polarization radar 4540 and a server device 1110 (an example of an estimation device).

[0252] The dual-polarized radar 4540 measures the rainfall space 4530 and obtains observations effective for estimating the particle size distribution (DSD), which is a physical variable related to the amount of precipitation. ｔ As the horizontally polarized radar reflectivity factor Z Ｈ (z ｔ ), and the difference in reflectivity factor between polarizations Z ｄｒ (z ｔ ), ・Phase difference between polarizations Φ ＤＰｔ , and outputs the observation value o measured by the dual polarization radar 4540. ｔ－Ｎ ~o ｔ＋Ｎ is transmitted to the server device 1110.

[0253] The server device 1110 functions as a posterior distribution estimator 4220. The posterior distribution estimator 4220 has a parameter-optimized smoothing distribution (first approximation model). The posterior distribution estimator 4220 estimates the observed value o ｔ－Ｎ ~o ｔ＋Ｎ Based on this, the posterior distribution q(z ｔ |o ｔ－Ｎ：ｔ＋Ｎ ) is estimated.

[0254] <Summary> As is clear from the above explanation, by applying the posterior distribution estimation technique described in the fifth embodiment to the estimation of the posterior distribution of the parameters of the raindrop particle size distribution in the rainfall space, according to the sixth embodiment, it becomes possible to accurately calculate the amount of precipitation.

[0255] [Other Embodiments] In this specification (including the claims), when the expression "at least one of a, b, and c" or "at least one of a, b, or c" (including similar expressions) is used, it includes any of a, b, c, a-b, a-c, bc, or a-bc. It may also include multiple instances of any element, such as a-a, a-bb-b, a-a-bb-cc-c, etc. Furthermore, it also includes adding elements other than the enumerated elements (a, b, and c), such as having d, as in a-b-c-d.

[0256] Furthermore, in this specification (including claims), when expressions such as "using data as input / based on / according to / in response to" (including similar expressions) are used, unless otherwise specified, this includes cases where various data itself is used as input, or where various data that has been processed in some way (e.g., noise-added, normalized, intermediate representation of various data, etc.) is used as input. Furthermore, when it is stated that a result is obtained "based on / according to / in response to data," this includes cases where the result is obtained based solely on the data in question, as well as cases where the result is obtained in response to other data, factors, conditions, and / or states other than the data in question. Furthermore, when it is stated that "data is output," unless otherwise specified, this includes cases where various data itself is used as output, or where various data that has been processed in some way (e.g., noise-added, normalized, intermediate representation of various data, etc.) is output.

[0257] Furthermore, when the terms "connected" and "coupled" are used in this specification (including the claims), they are intended as open-ended terms that include any of direct connection / coupling, indirect connection / coupling, electrically connection / coupling, communicatively connection / coupling, functionally connection / coupling, and physically connection / coupling. These terms should be interpreted appropriately depending on the context in which they are used, but any connection / coupling form that is not intentionally or naturally excluded should be interpreted as being included in these terms without any restrictions.

[0258] Furthermore, in this specification (including the claims), when the expression "A configured to B" is used, it may include the physical structure of element A having a configuration capable of performing operation B, and the permanent or temporary setting / configuration of element A being configured / set to actually perform operation B. For example, if element A is a general-purpose processor, it is sufficient that the processor has a hardware configuration capable of performing operation B, and is configured to actually perform operation B by setting a permanent or temporary program (instruction). Furthermore, if element A is a dedicated processor or dedicated arithmetic circuit, it is sufficient that the circuit structure of the processor is implemented to actually perform operation B, regardless of whether control instructions and data are actually attached.

[0259] Furthermore, when words implying containing or possessing (e.g., "comprising / including" and "having") are used in this specification (including the claims), they are intended to be open-ended terms that include cases where something other than the object indicated by the object of the term is contained or possessed. When the object of such words implying containing or possessing does not specify a quantity or suggests a singular number (e.g., an expression using the article "a" or "an"), the expression should be construed as not being limited to a specific number.

[0260] Furthermore, although expressions such as "one or more" or "at least one" are used in some places in this specification (including the claims) and expressions that do not specify a quantity or suggest a singular number (expressions using the articles "a" or "an") are used in other places, the latter expressions are not intended to mean "one." In general, expressions that do not specify a quantity or suggest a singular number (expressions using the articles "a" or "an") should be interpreted as not necessarily being limited to a specific number.

[0261] Furthermore, if a particular advantage / result is described in this specification as being obtained with respect to a particular configuration of an embodiment, it should be understood that the same advantage / result can also be obtained with one or more other embodiments having the same configuration, unless otherwise stated. However, it should be understood that the presence or absence of the effect generally depends on various factors, conditions, and / or states, etc., and that the effect is not necessarily obtained with the configuration. The effect is merely obtained by the configuration described in the embodiment when various factors, conditions, and / or states, etc. are satisfied, and the effect does not necessarily occur in a claimed invention that defines the same or a similar configuration.

[0262] Furthermore, in this specification (including claims), when multiple pieces of hardware perform a predetermined process, the pieces of hardware may cooperate to perform the predetermined process, or some of the hardware may perform all of the predetermined process. Furthermore, some of the hardware may perform part of the predetermined process, and other hardware may perform the rest of the predetermined process. In this specification (including claims), when an expression such as "one or more pieces of hardware perform a first process, and the one or more pieces of hardware perform a second process" is used, the hardware performing the first process and the hardware performing the second process may be the same or different. In other words, it is sufficient that the hardware performing the first process and the hardware performing the second process are included in the one or more pieces of hardware. Note that hardware may include an electronic circuit, a device including an electronic circuit, etc.

[0263] Furthermore, in this specification (including the claims), when multiple storage devices (memories) store data, each of the multiple storage devices (memories) may store only a portion of the data, or may store the entire data.

[0264] Although the embodiments of the present disclosure have been described in detail above, the present disclosure is not limited to the individual embodiments described above. Various additions, modifications, substitutions, partial deletions, etc. are possible within the scope of the conceptual idea and spirit of the present invention derived from the content defined in the claims and their equivalents. For example, in all of the above-described embodiments, the numerical values ​​used in the explanations are shown as examples and are not limited to these. Furthermore, the order of each operation in the embodiments is shown as an example and is not limited to these.

[0265] This application claims priority to U.S. Provisional Application No. 63 / 627,963, filed February 1, 2024, U.S. Provisional Application No. 63 / 669,327, filed July 10, 2024, and U.S. Provisional Application No. 63 / 681,278, filed August 9, 2024, the entire contents of which are incorporated herein by reference.

Claims

1. An estimation device comprising at least one memory and at least one processor, wherein, in the case of a nonlinear state space model in which the distribution of a state at a second time point evolved over time based on a state at a first time point is described by a linear Gaussian distribution and the distribution of observation values at the second time point is described by a nonlinear Gaussian distribution based on the state at the second time point, when estimating the posterior distribution of the state at the second time point based on the observation values up to the second time point, the at least one processor calculates the posterior distribution of the state at the second time point based on the observation values up to the second time point using a model that estimates a linear Gaussian distribution, the model having parameters optimized to maximize the lower bound of the log-likelihood.

2. The estimation device according to claim 1, wherein the model includes an inverse observation operator that estimates a distribution of states at the second time point based on observed values at the second time point.

3. The estimation device according to claim 2, wherein the model calculates a posterior distribution of the state at the second time point based on the observation values up to the second time point using a distribution of the state at the second time point estimated based on the observation values up to the first time point and a distribution of the state at the second time point estimated based on the observation values at the second time point using the inverse observation operator.

4. The estimation device according to claim 3, wherein the state space model includes a dynamics model in which the distribution of the state at the second time point that evolves over time based on the state at the first time point is described by a nonlinear non-Gaussian distribution, by introducing a latent variable into the dynamics model.

5. The estimation device according to claim 4, wherein the model includes an inverse observation operator that estimates the distribution of the latent variables at the second time point based on the observed values at the second time point.

6. The estimation device according to claim 5, wherein the model calculates a posterior distribution of the latent variables at the second time point based on the observed values up to the second time point using a distribution of the latent variables at the second time point estimated based on the observed values up to the first time point and a distribution of the latent variables at the second time point estimated based on the observed values at the second time point using the inverse observation operator.

7. The estimation device according to claim 4, wherein, when parameters of a dynamics model described by a nonlinear non-Gaussian distribution are known and parameters of an observation model in which a distribution of observation values at the second time point is described by a nonlinear Gaussian distribution based on the state at the second time point are known, the parameters of the model are optimized using a combination of simulated values of the state at each time point and simulated values of the observation values at each time point obtained by recursively calculating a prediction step of executing the dynamics model described by the nonlinear non-Gaussian distribution and an update step of executing the observation model described by the nonlinear Gaussian distribution.

8. The estimation device according to claim 3, wherein the state space model includes a dynamics model described by a linear Gaussian distribution, obtained by transforming, using a Koopman operator, a dynamics model in which a distribution of a state at the second time point that has evolved over time based on the state at the first time point is described by a nonlinear Gaussian distribution.

9. The estimation device according to claim 8, wherein, when parameters of a dynamics model described by the nonlinear Gaussian distribution are known and parameters of an observation model in which a distribution of observation values at the second time point is described by a nonlinear Gaussian distribution based on the state at the second time point are known, the parameters of the model are optimized using a combination of simulated values of the state at each time point and simulated values of the observation values at each time point obtained by recursively calculating a prediction step of executing the dynamics model described by the nonlinear Gaussian distribution and an update step of executing the observation model described by the nonlinear Gaussian distribution.

10. The estimation device according to claim 3, wherein the state space model includes a dynamics model described by a linear Gaussian distribution, obtained by Taylor expansion of a dynamics model in which the distribution of the state at the second time point evolved over time based on the state at the first time point is described by a nonlinear Gaussian distribution.

11. The estimation device according to claim 10, wherein the parameters of the model are optimized using observed values at each time point.

12. The estimation device according to claim 3, wherein the state space model includes a dynamics model in which a distribution of states at the second time point that evolve over time based on the states at the first time point is described by a linear Gaussian distribution.

13. The estimation device according to claim 12, wherein the parameters of the model are optimized using observed values at each time point.

14. An estimation device comprising at least one memory and at least one processor, wherein, when using a state space model to estimate a posterior distribution of a state at a given time point based on observation values up to a given time point, the at least one processor calculates the posterior distribution of the state at the given time point based on observation values local in the time direction for the given time point or observation values local in the space direction for a given measurement point at the given time point using a function that is invariant regardless of time point and spatial position, and calculates the posterior distribution of the state at the given time point using the model with optimized parameters under the assumption that the distribution of the state at the given time point has a correlation calculated by multiplying a constant attenuation coefficient for time intervals regardless of time point, or under the assumption that the distribution of the state at a given measurement point at the given time point has a correlation calculated by multiplying a constant attenuation coefficient for spatial intervals regardless of spatial position.

15. The estimation device according to claim 14, wherein the model is a first model that calculates a posterior distribution of the state at a given time based on observed values in a local time range before and after the given time or observed values in a local spatial range around a given measurement point at the given time, and whose parameters are optimized to maximize a lower bound on the log-likelihood.

16. The estimation device according to claim 15, wherein the model is a second model that calculates a posterior distribution of the state at a given time based on observed values in a local time range up to the given time point or observed values in a local spatial range up to a given measurement point at the given time point, and the second model has parameters optimized to minimize the KL distance with a function including the first model.

17. An estimation method in which, in the case of a nonlinear state space model in which the distribution of states at a second time point evolved over time based on states at a first time point is described by a linear Gaussian distribution and the distribution of observed values at the second time point is described by a nonlinear Gaussian distribution based on states at the second time point, when estimating the posterior distribution of states at the second time point based on observed values up to the second time point, at least one processor calculates the posterior distribution of states at the second time point based on observed values up to the second time point using a model that estimates a linear Gaussian distribution, the model having parameters optimized to maximize the lower bound of the log-likelihood.

18. An estimation method in which, when using a state space model to estimate the posterior distribution of a state at a given time based on observation values up to a given time, at least one processor calculates the posterior distribution of the state at the given time based on observation values local in the time direction for the given time or observation values local in the space direction for a given measurement point at the given time using a function that is invariant regardless of time and spatial position, and calculates the posterior distribution of the state at the given time using the model with optimized parameters under the assumption that the distribution of the state at the given time has a correlation calculated by multiplying a constant attenuation coefficient for time intervals regardless of time, or under the assumption that the distribution of the state at a given measurement point at the given time has a correlation calculated by multiplying a constant attenuation coefficient for spatial intervals regardless of spatial position.

19. In the case of a nonlinear state space model in which the distribution of states at a second time point evolved over time based on states at a first time point is described by a linear Gaussian distribution, and the distribution of observed values at the second time point is described by a nonlinear Gaussian distribution based on states at the second time point, an estimation program for causing at least one processor to execute processing to estimate the posterior distribution of states at the second time point based on observed values up to the second time point, using a model that estimates a linear Gaussian distribution, the parameters of which are optimized to maximize the lower limit of the log-likelihood.

20. An estimation program for causing at least one processor to execute a process of estimating the posterior distribution of a state at a given time point based on observation values up to the given time point using a state space model, the process comprising: calculating the posterior distribution of the state at the given time point based on observation values local in the time direction for the given time point or observation values local in the space direction for a given measurement point at the given time point using a function that is invariant regardless of time point and spatial position, and calculating the posterior distribution of the state at the given time point using the model with optimized parameters under the assumption that the distribution of the state at the given time point has a correlation calculated by multiplying a constant attenuation coefficient for time intervals regardless of time point, or under the assumption that the distribution of the state at a given measurement point at the given time point has a correlation calculated by multiplying a constant attenuation coefficient for spatial intervals regardless of spatial position.

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