Analysis of the application of Bayesian estimates as part of practical problems in dependability that involve models intended for groups of homogeneous products

. Experts in applied dependability are showing an increasing interest in the Bayesian theory. However, the Bayesian approach is not generally accepted by mathematical statistics and dependability researchers. The doubts about its practical applicability are primarily due to the fact that it allows for subjective probabilities. In practice, homogeneous product models are normally considered, i.e., each item in the evaluated batch is characterized by the same selected dependability value. In the case of Bayesian statistical estimation, the model involves heterogeneous products, however, in the course of steady production, it is not considered normal to manufacture products with varied dependability, which calls into question the adequacy of Bayesian statistical estimation methods.

Aim. It is demonstrated that using Bayesian estimators that employ models designed for homogeneous products in dependability is erroneous.

Methods of research. For the purpose of finding effective estimates within a selected class, integral numerical characteristics of the accuracy of estimation were used, namely, total squared bias of the expected implementation of a certain variant estimate from the examined parameters of the distribution laws, etc.

Conclusions. 1. For any result, the realizations of Bayesian estimates are grouped within the dogma of mean – ≈p_α = 1 – α / (α + β), while classical ^≈Р₂ = 1 – R/N  and integral ^≈Р₄ unbiased estimates respond adequately to any external changes. The use of Bayesian estimates in dependability, when models designed for homogeneous products are employed, is erroneous, and there is no need to use them. 2. Bayesian estimates should only be used for groups of heterogeneous products. 3. Instead of Bayesian estimates, integral biased estimates should be used in dependability, when models designed for homogeneous products are employed.

1. Savchuk VP. [Bayesian methods of statistical evaluation: Dependability of technical facilities]. Moscow: Nauka; 1989. (in Russ.)

2. GOST RO 1410-001-2009. [Space systems. Procedure for dependability requirements definition, estimation and supervision]. Moscow: Standartinform; 2015. (in Russ.)

3. Kolobov A.Yu., Dikoun E.V. Interval estimation of reliability of one-off spacecraft. Dependability 2017;4:23-26.

4. Borovkov AA. [Mathematical statistics]. Novosibirsk: Nauka; 1997. (in Russ.)

5. Shulenin V.P. [Mathematical statistics. Part 2. Nonparametric statistics]. Tomsk: Izdatelstvo NTL; 2012. (in Russ.)

6. Krupkina T.V. [Probability theory and mathematical statistics. Part 2. Electronic series of lectures]. Krasnoyarsk: Siberian Federal University; 2011. (in Russ.)

7. Mikhaylov V.S. [Study of estimates based on the integral and Bayesian methods]. Reliability & Quality of Complex Systems 2018;1(21):28-39. (in Russ.)

8. Mikhaylov V.S., Yurkov N.K. [Composite Bayesian estimates]. Reliability & Quality of Complex Systems 2019;4(28):118-126. (in Russ.)

9. Mikhailov V.S., Yurkov N.K. [Integral estimation in the dependability theory. Introduction and main findings]. TEKHNOSFERA; 2020. (in Russ.)

10. Mikhailov V.S. Efficiency criterion of biased estimates. A new take on old problems. Dependability 2022;1:30-37.

11. Gnedenko B.V., Beliaev Yu.K., Soloviev A.D. [Mathematical methods in the dependability theory]. Moscow: Nauka; 1965. (in Russ.)