Article 13419

Title of the article

A COMPOSITE BAYESIAN ESTIMATE 

Authors

Mikhailov Viktor Sergeevich, lead engineer, Central Research Institute of Chemistry and Mechanics named after D. I. Mendeleev (115487, 16а Nagatinskaya street, Moscow, Russia), E-mail: Mvs1956@list.ru
Yurkov Nikolay Kondratievich, doctor of technical sciences, professor, the honoured worker of science of the Russian Federation, head of sub-department of radio equipment design and production, Penza State University (440026, 40 Krasnaya street, Penza, Russia), E-mail: yurkov_NK@mail.ru 

Index UDK

519.248:62-192 

DOI

10.21685/2307-4205-2019-4-13 

Abstract

Background. The purpose of the article is to determine the rules for constructing a composite Bayesian estimate. Modeling various situations of the world should be aware of its constant variability. Thus, different batches of products have different values of the parameters of the a priori distribution, and in case of violation of technological discipline, these differences become even stronger. However, to notice these differences does not allow not only the fixed choice of the type of a priori distribution, but also the fixed choice of the values of the parameters of this distribution, made on samples of different batches of products before the release of the controlled batch. Those. This choice is based on the experience of a stable release of previous batches. So The Bayesian estimate of the controlled lot directly (strongly, sensitively) depends not only on the type of a priori distribution, the outcome of the tests and the sample size N, but also on the selected parameters of the a priori distribution. In practice, the most common case is the two parametric prior distribution. Therefore, the further presentation, without disturbing the generality of the reasoning, will be carried out for the two parametric case q (t, α, β). The problem of inadequate response to changes in the process of the controlled batch of products can be solved if the Bayesian estimate of the reliability index of each controlled batch of products is sought as a composite estimate characterized by a priori established distribution (or established distributions), the parameters of which are predetermined depending on the results future test products of this batch.
Those. The parameters α and β should not determine the entire technological process, stretching
in time, but only a controlled batch of products depending on the test result. For the example described, the Bayesian estimate should be represented as θ ̂ Q (R, N, α_i, β_i), where i = 0, 1, 2, ..., N and with R = r, i = r; and the resulting density of the composite a priori beta distribution is in the form of q (t, α_i, β_i), where α_i, β_i is a set of parameters, which is calculated from the results of tests of a controlled batch of products in accordance with the formulated rules. This model will allow to determine the real changes, not the stability of the situation, based on the results of a composite Bayesian assessment. It is clear that the selection of the parameters α_i, β_i depends on the specific test plan and the probability of occurrence of the test outcome P (r, N, p), which is necessary to choose the maximum P_max = P (r, N, p_max) for this test plan and outcome in order to obtain estimates p ̂ = p_max of the parameter p, which simultaneously determines the a priori estimate of the achieved level of reliability of the controlled batch p ̂ α, i.e. p ̂ = p ̂ α = α_r / (α_r + β_r). Note that a priori, the known distribution density should not be obtained from the results of tests of various batches of products, but from the known dependencies of the probability of occurrence of outcomes P (r, N, p) for a specific test plan, which eliminates errors depending on the properties of the used statistical evaluation of the desired parameters α , β in the classic case. Note once again that the parameters α_r, β_r are not pure a priori, but are selected options from a predetermined set of parameters depending on the results of testing a test batch of products. The pair of parameters α_r, β_r determined in such a way, from an already defined set of options determined by the rule of maximum probability of occurrence of events (failures) of a specific test plan, characterizes the reliability of a controlled batch of products. A priori information can be considered the distribution itself (for example, the beta distribution of the parameter p, the randomized parameter of the binomial distribution). The composite Bayes score does not depend on the type (type) of the tested products, but depends only on the known
(proven) dependence of the probability of the test outcome, which for a particular test plan, due to the proof of the choice of probability according to the maximization rules, this probability does not change. This is the main advantage of the composite Bayesian estimate – a set of options for the parameters α_r, β_r is determined (calculated) in advance for a specific test plan once and before testing. It remains only to determine the specific values of these parameters for a controlled batch of products based on the results of tests on the basis of an established set of options for the parameters α_r, β_r. 

Key words

beta distribution; binomial distribution law; test plan; point estimation; Bayesian estimation 

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Дата создания: 03.02.2020 16:33
Дата обновления: 04.02.2020 11:14