Bayesian Uncertainty Evaluation Method Based on Acceptance-Rejection Sampling
Through the Bayesian uncertainty assessment method based on acceptance-reject sampling, the problem of difficulty in obtaining posterior distribution in Bayesian method is solved, and a more accurate and reliable measurement uncertainty assessment is achieved, especially when fusing a priori information, the evaluation process of complex measurement models is simplified.
Patent Information
- Application Number
- CN202310973476.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-08-03
- Publication Date
- 2025-07-29
- Estimated Expiration
- 2043-08-03
AI Technical Summary
The prior art faces the problem of difficulty in obtaining posterior distributions when Bayesian methods perform uncertainty assessment, especially when fusion of prior information, it is difficult to achieve effective statistical inference.
The Bayesian uncertainty assessment method based on acceptance-reject sampling is adopted. By establishing a measurement model, determining a priori distribution, using the Monte Carlo method for sampling, and obtaining posterior samples in combination with the acceptance-reject sampling algorithm, the acquisition process of posterior distribution is simplified.
A more comprehensive and accurate measurement uncertainty assessment is achieved, especially under the conditions of historical information, the measurement results are more reliable, simplifying the evaluation process of complex measurement models.
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Figure CN117114115B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of instrument measurement result evaluation methods, and specifically to a Bayesian uncertainty evaluation method based on acceptance-rejection sampling. Background Art
[0002] Measurement uncertainty evaluation is a process of quantitatively estimating the uncertainty of measurement results, which is of great significance in science, engineering, and experimental research. For example, evaluating the measurement uncertainty of the measurement results of a measuring instrument can help determine the reliability and accuracy of the measurement results. By calculating the uncertainty, one can understand the credibility of the measurement results and know whether the accuracy of the experiment or measurement is sufficient to meet the requirements of research or engineering applications. In addition, when comparing different experimental or measurement results, the uncertainty needs to be considered. Only when the uncertainty ranges of the measurement results overlap can the two results be considered consistent or similar. By evaluating the uncertainty, data verification and comparison can be carried out to ensure the repeatability and accuracy of the research. Moreover, in many practical applications, decisions need to rely on measurement results. Uncertainty evaluation provides information related to decision-making, enabling decision-makers to understand the credibility of the measurement results and make reasonable decisions. For scientific research or engineering applications, accurate data interpretation is crucial. Uncertainty evaluation can help explain the range and confidence interval of the measurement results, making the results easier to interpret and understand. In experimental research, evaluating the measurement uncertainty can help optimize the experimental design, thereby reducing unnecessary measurement times or resource waste and improving experimental efficiency. Research and applications in many fields need to meet certain standards and specifications. Measurement uncertainty evaluation is an important means to ensure that the research or application results meet the standards and specifications.
[0003] Therefore, measurement uncertainty evaluation is of great significance in ensuring data quality, improving experimental accuracy, supporting scientific decision-making, and optimizing research design, etc. It can promote the understanding of the uncertainty of measurement results, provide support for data interpretation and application, and is an indispensable part of scientific research and engineering applications.
[0004] At present, two uncertainty evaluation methods provided by the "Guide to the Expression of Uncertainty in Measurement" (GUM) issued by international organizations and its supplementary document GUM-S1 have been applied throughout metrology. Among them, the uncertainty evaluation method of the GUM-S1 method is applicable not only to linear measurement models but also to non-linear measurement models or cases where the measured quantity conforms to a non-Gaussian distribution. It adopts the strategy of Monte Carlo sampling in the process of uncertainty evaluation. However, both the GUM and GUM-S1 methods are based on existing data for measurement uncertainty evaluation, ignoring the prior information about the measured quantity. In actual measurement work, some prior information about the measured quantity can usually be obtained, such as measurement results obtained under the same environmental conditions before, results measured by more advanced precision instruments, empirical data given by relevant experts, etc. These prior information of the measured quantity contain some information factors of the measured part itself. Using Bayesian statistical thinking, the prior information and sample information can be fused into the latest data information. Therefore, Bayesian statistical thinking is applied to the process of measurement uncertainty evaluation.
[0005] However, when actually using the Bayesian method for uncertainty evaluation, some difficulties are often faced in the evaluation process, mainly manifested as the difficulty in obtaining the posterior distribution of the measured quantity. If the posterior distribution is directly obtained by integration through the Bayesian formula, the general solution process is very difficult. To avoid complex integral solutions, the Markov Chain Monte Carlo (MCMC) method can be used to sample the posterior distribution, but it is still difficult to determine whether the sampling has reached a stationary state. These problems will affect the subsequent statistical inference of the uncertainty of the measured quantity. Summary of the Invention
[0006] The object of the present invention is to provide a Bayesian uncertainty evaluation method based on acceptance-rejection sampling to solve the problem of difficulty in obtaining the posterior distribution of the measured quantity in the evaluation process when using the Bayesian method for uncertainty evaluation.
[0007] To achieve the above object, the technical solution adopted by the present invention is as follows:
[0008] The Bayesian uncertainty evaluation method based on acceptance-rejection sampling includes the following steps:
[0009] Step 1: Based on the measured quantity, type B information input during the measurement process, and measurement data generated during the measurement process, establish a measurement model , where:
[0010] Y is the measured quantity; X is the directly measured quantity during the measurement process; B is the comprehensive representation of multiple type B information; f ( ) is based on the measured quantity, BMathematical function for determining the physical relationship between class parameters and measurement data;
[0011] Convert the measurement model to obtain a model , where g( ) is f the inverse function of ( );
[0012] Step 2: Determine the prior distribution of the measured quantity under the condition of no prior information or the prior distribution under the condition of historical information , the combined prior distribution of type B information ;
[0013] Step 3: Combine the information of the prior distribution obtained in Step 2 and the information of the prior distribution to sample the model using the Monte Carlo method to obtain sampling points , where n represents the number of samplings by the Monte Carlo method, and the sampling point is the prior distribution information of the measurement data X , and thus obtain the prior distribution X of the measurement data ;
[0014] Step 4: Obtain the marginal likelihood function X of the model from the measurement data , where represents the measurement data generated by multiple independent measurements during the measurement process;
[0015] Step 5: Use the acceptance-rejection sampling algorithm to obtain a random sampling value X i corresponding to the uniform distribution , where i = 1, 2,... n; then substitute the sampling point i into the marginal likelihood function obtained in Step 4 to get X , and then compare with the random sampling value obtained in Step 5. If , accept and retain the sampling point , otherwise reject and discard the sampling point ; .
[0016] Step 6: Use the accepted sampling points in Step 5 and retain the corresponding as the posterior samples of the measured quantity Y . Based on the posterior samples Y of the measured quantity Obtain the best estimate value, uncertainty, and coverage interval index of the measurand. Y
[0017] In the further step 2, there are two states of the prior distribution information of the measurand . When the measurand is obtained by the first measurement, the measurand has no historical information. At this time, the prior distribution is obtained according to the Bayesian hypothesis under the non-informative prior condition, that is, the prior distribution of the measurand is represented by a uniform distribution . The upper and lower limits of the uniform distribution X are obtained from the range of the measurement data after being transmitted through the measurement model;
[0018] When the measurand has historical information, the prior distribution of the measurand can be obtained by combining the distribution information formed by the measurement data and the B-type information to form a combined prior distribution and using the Monte Carlo method.
[0019] In the further step 3, substitute the prior distribution and the prior distribution obtained in step 2 into the model . Then, use the Monte Carlo method to sample the model to obtain the sampling points .
[0020] The process of the further step 4 is as follows:
[0021] In the process of using the Bayesian method in the model , is the sample likelihood function, and the sample likelihood function represents the probability of obtaining the measurement data . Assuming Bayesian uncertainty evaluation is carried out under the condition of non-informative data variance parameters, can be obtained according to the Jeffreys prior, that is, the corresponding marginal likelihood function is obtained as follows under the condition of non-informative data variance parameters:
[0022]
[0023] where: the value of the measurement data X is known from the model ;
[0024] is the variance parameter for the data following a normal distribution;
[0025] denotes the Jeffreys prior distribution of
[0026] , s respectively denote the mean and variance of the measurement data generated during the measurement process where the mean is , and the variance is .
[0027] The further process of Step 5 is as follows:
[0028] First, set a probability distribution function convenient for sampling , and then set a constant to make the known probability density function always below , satisfying ;
[0029] Then, randomly sample a value from the uniform distribution . Since , when takes the value of 1, that is, at this time, the sampling value is randomly obtained from the uniform distribution ;
[0030] Then, substitute the sampling point X i into the marginal likelihood function obtained in Step 4 to get , and then compare with the random sampling value . If , accept and retain the sampling point , otherwise reject and discard the sampling point .
[0031] The probability distribution function in the further Step 5 is a Gaussian distribution or a uniform distribution.
[0032] In the further Step 6, the posterior samples are sorted in ascending order to obtain the distribution function of the measured quantity Y . According to the distribution function of the measured quantity Y , the best estimate value, uncertainty, and coverage interval index of the measured quantity Y are obtained.
[0033] Compared with the prior art, the advantages of the present invention are:
[0034] (1) The present invention applies Bayesian statistical thinking to the process of measuring uncertainty evaluation. By using Bayesian hypothesis or Monte Carlo method to obtain the prior information of the measured quantity, the prior information and sample information are fused into the latest data information, and more comprehensive and accurate measurement uncertainty can be obtained.
[0035] (2) The present invention realizes the Bayesian uncertainty evaluation method based on the acceptance - rejection sampling algorithm, simplifies the process of obtaining the Bayesian posterior distribution, and is conducive to the Bayesian uncertainty evaluation for more complex measurement models.
[0036] (3) The best estimated values of the measurement results obtained by the present invention are basically the same as those obtained by GUM and MCM. Under the condition of no information about the measured quantity, the standard uncertainty of the measured quantity obtained is basically the same as the evaluation results of GUM and MCM; under the condition of having information about the measured quantity, the standard uncertainty of the measured quantity obtained is smaller than the evaluation results of GUM and MCM. Due to the fusion of historical prior information, the measurement results are more reliable. This method can provide a method reference for realizing the measurement uncertainty evaluation for complex tasks. Description of the Drawings
[0037] Figure 1 is the flowchart of the Bayesian method for uncertainty evaluation based on acceptance - rejection sampling in the embodiment of the present invention;
[0038] Figure 2 is the information table of the input quantity of the quality calibration model in the embodiment of the present invention;
[0039] Figure 3 is the sample information table of the measurement parameter in the embodiment of the present invention;
[0040] Figure 4 is the probability density distribution diagram of the evaluation results of the GUM method, MCM method and Bayesian method based on acceptance - rejection sampling under the condition that the measured quantity has no history in the embodiment of the present invention;
[0041] Figure 5 is the probability density distribution diagram of the evaluation results of the GUM method, MCM method and Bayesian method based on acceptance - rejection sampling under the condition that the measured quantity has history in the embodiment of the present invention. Detailed Embodiments
[0042] The present invention will be further described below in conjunction with the drawings and embodiments.
[0043] As Figure 1 shown, this embodiment discloses a Bayesian uncertainty evaluation method based on acceptance - rejection sampling, including the following steps:
[0044] Step 1: Based on the measured quantity, type - B information input during the measurement process, and measurement data generated during the measurement process, establish a measurement model , where:
[0045] Y Y is the quantity to be measured; X X is the directly measured quantity in the measurement process; B Z is the comprehensive representation of multiple Class B information; f f( ) is a mathematical function determined according to the physical relationship between the quantity to be measured, B Class B parameters, and measurement data.
[0046] The measurement model is transformed to obtain model , where g( ) is f the inverse function of f( ).
[0047] This embodiment is described by taking the measurement of the mass of a calibration weight as an example. During the measurement process, a reference weight with a nominal mass is used for measurement, and the air density, air mass density, mass density of the weight to be calibrated, and mass density of the reference weight are considered. According to the physical relationship between the quantity to be measured, measurement data, and the input Class B information, the established measurement model is shown in formula (1):
[0048] (1),
[0049] where:
[0050] mR is the reduced mass of the calibration reference weight R, ρ0 is the density of the air in which the counterweight is balanced at a density of ( ) and the density of the reference weight is mR, m0 is the nominal mass and , mW is the reduced mass of the weight W to be calibrated and its deviation from the nominal mass m0, that is Y is the quantity to be measured.
[0051] ρa is the air mass density. ρW is the mass density of the weight to be calibrated. ρR is the mass density of the reference weight. Class B information refers to the information obtained through methods such as the calibration certificate of the instrument, historical records, literature materials, or expert knowledge. In this embodiment, each Class B information is as shown in Figure 2 , including the reduced mass mR of the reference weight R, ρa of the air mass density, ρW of the mass density of the weight to be calibrated, ρR of the mass density of the reference weight.
[0052] The reduced mass of the small weights with density added to the reference weight R to achieve balance with the weight W to be calibrated during calibration. For n independent measurements, the measurement data obtained in this example are as shown in Figure 3 .
[0053] The measurement model shown in formula (1) is subjected to an inverse function transformation to obtain the model , that is, the model as shown in formula (2):
[0054] (2).
[0055] Step 2. Determine the prior distribution of the measured quantity (i.e., the deviation ) under the condition of no prior information or the prior distribution under the condition of historical information = , the combined prior distribution of type B information . Among them:
[0056] (2.1) There are two states of the prior distribution information of the measured quantity . When is obtained from the first measurement, there is no historical information. At this time, the prior distribution is obtained according to the Bayesian hypothesis under the condition of non-informative prior, that is, the prior distribution of is represented by a uniform distribution [[ID= forty-eight]]The upper and lower limits of the uniform distribution are obtained from the range of the measurement data and the intervals of each type B information after being transmitted through the measurement model .
[0057] When is in the state of having historical information, at this time, the prior distribution of the measured quantity can be obtained by combining the distribution information formed by the measurement data and the combined prior distribution of type B information and using the Monte Carlo method.
[0058] That is, the data containing the measured values also contains type B information, which is to a certain extent the basic range of the posterior samples of the measured quantity to be obtained.
[0059] (2.2) In step 2, the combined prior distribution of type B information is obtained through methods such as the calibration certificate of the instrument, historical records, literature materials, or expert knowledge.
[0060] Step 3. Then, according to the prior distribution obtained in Step 2 = information, the prior distribution information, after the prior distribution and the prior distribution are substituted into the model shown in Formula (2) , and then the model shown in Formula (2) is sampled using the Monte Carlo method to obtain sampling points , where n represents the number of times of sampling by the Monte Carlo method. The sampling points are the prior distribution information of the measurement data X (i.e., the reduced mass ), and thus the prior distribution is obtained .
[0061] Step 4. According to the measurement data X (i.e., the reduced mass ), the marginal likelihood function of the model shown in Formula (2) is obtained as follows:
[0062] In the Bayesian process using the model shown in Formula (2) , is the sample likelihood function, and the sample likelihood function represents the probability of obtaining the measurement data . Assuming Bayesian uncertainty evaluation is carried out under the condition of no information on the data variance parameter, can be obtained according to the Jeffreys prior, that is, the corresponding marginal likelihood function is obtained as follows under the condition of no information on the data variance parameter:
[0063] ,
[0064] where: The value of is known from the model shown in Formula (2);
[0065] According to the measurement data shown in Figure 3 , it is known from the model shown in Formula (2) that , is the variance parameter for the data to follow a normal distribution; represents the measurement data generated by multiple measurements of during the measurement process;
[0066] represents the Jeffreys prior distribution of
[0067] , s respectively represent the mean and variance of the measurement data where the mean is and the variance is .
[0068] Step 5: Use the acceptance-rejection sampling algorithm to obtain any sampling point X i The corresponding random sampling value on the uniform distribution where i = 1, 2, …… n, and the process is as follows:
[0069] First, set a probability distribution function convenient for sampling , and this probability distribution function uses a Gaussian distribution function or a uniform distribution function.
[0070] Then set a constant to make the known probability density function always below and satisfy .
[0071] Then randomly sample a value from the uniform distribution . Since , when takes the value of 1, that is, at this time, a random sampling value is randomly obtained on the uniform distribution ;
[0072] Then substitute the sampling point X i into the marginal likelihood function in Step 4 to obtain , and then compare with the random sampling value obtained in Step 5. If , accept and retain the sampling point , otherwise reject and discard the sampling point .
[0073] Step 6: Use the sampling points accepted and retained in Step 5 when using the Monte Carlo method to sample the prior distribution as the posterior samples of . After sorting the posterior samples in ascending order, obtain the distribution function of . According to the distribution function of , obtain . According to the distribution function of Best estimate values, uncertainties, and coverage interval indicators. Specifically, according to the formulas in the Monte Carlo method the best estimate value is obtained, and according to the formula in the Monte Carlo method the standard uncertainty is obtained, and according to the Monte Carlo method the coverage interval is obtained, where for any , it satisfies , , and is the confidence probability, is the number of Monte Carlo samplings. If is an integer, let , otherwise take as the integer part.
[0074] Figure 4 and Figure 5 are the comparisons between the results of this embodiment and the prior art. It can be seen from Figure 4 that this embodiment can obtain a best estimate value of the measurement result that is basically the same as that obtained by the GUM and MCM evaluation methods; under the condition that there is no information about the measured quantity (i.e., the output quantity in Figure 4 , Figure 5 ), the standard uncertainty of the measured quantity obtained by the evaluation method of this embodiment is basically the same as that of the GUM and MCM evaluation results. It can be seen from Figure 5 that this embodiment can obtain a best estimate value of the measurement result that is basically the same as that obtained by the GUM and MCM evaluation methods; under the condition that there is historical information about the measured quantity, the standard uncertainty of the measured quantity obtained by the evaluation method of this embodiment is smaller than that of the GUM and MCM evaluation results. Due to the integration of historical prior information, the measurement result is more reliable.
[0075] The present invention provides a Bayesian uncertainty evaluation method based on acceptance-rejection sampling to solve the problem of difficulty in obtaining the posterior distribution of the measurement model in Bayesian uncertainty evaluation. This method can be applied to measurement uncertainty evaluation.
[0076] The preferred embodiments of the present invention have been described in detail above with reference to the accompanying drawings. The embodiments described in the present invention are only descriptions of the preferred embodiments of the present invention, and do not limit the concept and scope of the present invention. Among the various specific technical features described in the above specific embodiments, they can be combined in any appropriate way without contradiction. As long as such a combination does not violate the idea of the present invention, it should also be regarded as the content disclosed in the present disclosure. To avoid unnecessary repetition, the present invention does not separately describe various possible combination methods.
[0077] The present invention is not limited to the specific details in the above embodiments. Without departing from the technical concept of the present invention and within the premise of not deviating from the design idea of the present invention, various modifications and improvements made by those skilled in the art to the technical solutions of the present invention shall fall within the protection scope of the present invention. The technical content claimed by the present invention has been fully recorded in the claims.
Claims
1. A Bayesian uncertainty evaluation method based on acceptance-rejection sampling, characterized in that, Including the following steps: Step 1: Establish a measurement model based on the measured quantity, type B information input during the measurement process, and measurement data generated during the measurement process , where: Y is the quantity to be measured; X is the directly measured quantity during the measurement process; B is the comprehensive representation of multiple Class B information; f ( ) is a mathematical function determined according to the physical relationship between the quantity to be measured, B Class parameters and measurement data; Convert the measurement model to obtain a model , where g( ) is f the inverse function of ( ); Step 2, determine the quantity to be measured The prior distribution under the condition of no prior information or the prior distribution under the condition of historical information , and the combined prior distribution of type B information ; Step 3: Combine the prior distribution information obtained in Step 2 and the prior distribution information to sample the model using the Monte Carlo method to obtain sampling points , where n represents the number of times of sampling by the Monte Carlo method, and the sampling points are the prior distribution information of the measurement data X . Thus, the prior distribution X of the measurement data is obtained; Step 4: According to the measurement data X obtain the marginal likelihood function of the model , where represents the measurement data generated by multiple independent measurements during the measurement process; Step 5: Use the acceptance-rejection sampling algorithm to obtain any sampling point X i at the corresponding random sampling value on the uniform distribution , where i = 1, 2, …… n; then substitute the sampling point X i into the marginal likelihood function obtained in Step 4 to get , and then compare with the random sampling value obtained in Step 5 . If , accept and retain the sampling point , otherwise reject and discard the sampling point ; Step 6: Using the sampling points accepted in Step 5 and retaining the corresponding as the posterior samples of the measured Y Based on the posterior samples of the measured Y the posterior samples obtain the best estimate value, uncertainty and coverage interval index of the measured Y ; The mass is measured as that of the weight. The type-B information includes the converted mass of the weight, the air density of the weight, the mass density of the weight, and the reference mass density of the weight.
2. The Bayesian uncertainty evaluation method based on acceptance-rejection sampling according to claim 1, wherein In step 2, the prior distribution information of the measured quantity has two states. When the measured quantity is obtained from the first measurement, the measured quantity has no historical information. At this time, the prior distribution is obtained according to the Bayesian hypothesis under the non-informative prior condition, that is, the prior distribution of the measured quantity is represented by a uniform distribution, and the upper and lower limits of the uniform distribution X are obtained from the range of the measurement data after being transmitted through the measurement model; When the measured quantity is in a state with historical information, the prior distribution of the measured quantity can be obtained by combining the distribution information formed from the measurement data with the type-B information to form a combined prior distribution and then using the Monte Carlo method. 3. The Bayesian uncertainty evaluation method based on acceptance-rejection sampling according to claim 1, wherein In step 3, the prior distribution obtained in step 2 , the prior distribution is substituted into the model . Then, the Monte Carlo method is used to sample the model to obtain the sampling points .
4. The Bayesian uncertainty evaluation method based on acceptance-rejection sampling according to claim 1, wherein The process of step 4 is as follows: In the model when using the Bayesian process, is the sample likelihood function, and the sample likelihood function represents the probability of the obtained measurement data . Assuming Bayesian uncertainty evaluation is carried out under the condition of uninformative data variance parameters, can be obtained according to the Jeffreys prior , that is, the corresponding marginal likelihood function is obtained under the condition of uninformative data variance parameters as follows: ， Among them: measurement data X The value of which can be known from the model , is the variance parameter for the data following a normal distribution; denote Jeffreys prior distribution; , s respectively represent the mean and variance of the measurement data generated during the measurement process , where the mean is , and the variance is .
5. The Bayesian uncertainty evaluation method based on acceptance-rejection sampling according to claim 1, wherein The process of step 5 is as follows: First, set a probability distribution function that is convenient for sampling. , and then set a constant to make the known probability density function always below , satisfying ; Randomly sample a value from the uniform distribution again . Since when takes the value 1, that is, at this time, a sampling value is randomly obtained from the uniform distribution again ; Then substitute the sampling points X i into the marginal likelihood function obtained in Step 4 to get . Then compare with the random sampling value . If , accept and retain the sampling point , otherwise reject and discard the sampling point .
6. The Bayesian uncertainty evaluation method based on acceptance-rejection sampling according to claim 5, wherein, The probability distribution function in step 5 is a Gaussian distribution or a uniform distribution.
7. The Bayesian uncertainty evaluation method based on acceptance-rejection sampling according to claim 1, characterized in that In step 6, for the posterior samples perform an ascending sort to obtain the measured Y distribution function. Based on the distribution function of the measured Y obtain the best estimate value, uncertainty, and coverage interval index of the measured Y .
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