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Multi-step algorithm for constructing statistical estimates based on the Bayesian approach in measuring tasks
https://doi.org/10.32446/0368-1025it.2022-4-23-29
Abstract
A multi-step algorithm based on the Bayesian approach to constructing effective statistical estimates of measurement results has been developed. A general logical scheme of a multi-step algorithm is presented, which allows, in the case of heterogeneity of a priori and a posteriori data, to build generalized Bayesian estimates based on the use of a mixture of a priori and a posteriori distributions. Conditions for the existence of a conjugate family of a priori distributions are formulated. A technique for calculating specific values of parameters in conjugate a priori distributions is described. The technology for applying the Bayesian approach is described in more detail for the binomial and negative-binomial distribution laws. Using the likelihood function, formulas for recalculating the parameters of the corresponding conjugate distribution law are obtained, which allow one to pass from the next step to the next step of the multi-step algorithm. Examples of applying the algorithm to the problem of assessing the compliance of measuring systems with specified requirements, as well as to the problem of processing the results of quantitative physical and chemical analysis are given. The presented results show that the Bayesian approach gives a significant gain in the accuracy of constructing statistical estimates for small and medium sample sizes. This circumstance makes the Bayesian approach particularly effective in the problems of assessing the metrological characteristics of measuring systems in the case when repeated repetition of tests is inexpedient or time consuming. On specific examples, it is illustrated that with an increase in the volume and number of samples, the multi-step Bayesian approach and the classical maximum likelihood method will give identical results.
About the Author
R. Z. Khayrullin
Moscow State University of Civil Engineering (National Research University);
Main Scientific Metrological Centre
Russian Federation
Russian Federation
Rustam Z. Khayrullin
Moscow,
Mytischy, Moscow region
References
1. Sychev E. I., Evaluation of the eff ectiveness and parametric synthesis of metrological support for radio equipment, Moscow, MO SSSR Publ., 1984. 386 p. (In Russ.)
2. Surdu M. N., Measurement Techniques, 2019, vol. 62, no. 6, pp. 562–569. https://doi.org/10.1007/s11018-019-01661-7
3. Kuznetsov V. A., Isaev L. K., Shaiko I. A., Metrology, Moscow, Standartinform, 2005, 298 p. (In Russ.)
4. Sychev E. I., Khramenkov V. N., Shkitin A. D., Fundamentals of metrology of military equipment, Moscow, Voennoe izdatel’stvo Publ., 1993, 400 p. (In Russ.)
5. Syusina O. M., Chernitsov A. M., Tamarov V. A., Solar System Research, 2012, vol. 46, no. 3, pp. 195–207. https://doi.org/10.1134/S0038094612020104
6. Vishnyakov B. V., Egorov A. I., Construction of confi dence regions for object motion trajectories in machine vision problems, Proceedings of the Russian Academy of Sciences. Theory and control systems, 2013, no. 3, pp. 124–132. (In Russ.)
7. Ayvazyan S. A., Bayesian approach in econometric analysis, Applied Econometrics, 2008, no. 1 (9). pp. 93–130. (In Russ.)
8. Ayvazyan S. A., Mkhitaryan V. S., Prikladnaya statistika i osnovy ekonometriki, in 2 volums, vol. 1, 2nd edition, corrected, Moscow, Uniti-Dana Publ., 2001, 656 p. (In Russ.)
9. Lancaster A., An Introduction to Modern Bayesian Econometrics, Blackwell Pub., 2004.
10. Ghosh S. K., Hajra S., Paek A., Jayaram M., Annual Review of Biochemistry, 2006, vol. 75, pp. 211–241. https://doi.org/10.1146/annurev.biochem.75.101304.124037
11. Tipping М., Sparse Bayesian Learning and the relevance vector machine, Journal of Machine Learning Research, 2001, no. 1, pp. 211–244.
12. Kropotov D., Vetrov D., Optimal Bayesian Classifi er with Arbitrary Gaussian Regularizer, Proc. of 7th Open German-Russian Workshop on Pattern Recognition and Image Understanding (OGRW-7-2007), Ettlingen, Germany, 20–25 Aug. 2007.
13. Kropotov D., Vetrov D., Machine Learning, Proceedings of
14. the Twenty-Fourth International Conference (ICML 2007), Corvallis, Oregon, USA, June 20–24, 2007. https://doi.org/10.1145/1273496.1273554
15. Vapnik V. N., Bounds on the Rate of Convergence of Learning Processes, In book: The Nature of Statistical Learning Theory. Statistics for Engineering and Information Science, Springer, New York, 2000, pp. 69–91. https://doi.org/10.1007/978-1-4757-3264-1_4
16. Maxwell Chickering D., Heckerman D., Meek C., A Bayesian approach to learning Bayesian networks with local structure. https://doi.org/10.48550/arXiv.1302.1528
17. Bidyuk P. I., Terentiev A. N., Construction and learning methods for Bayesian networks, Taurian Bulletin of Informatics and Mathematics, 2004, no. 2, pp. 1–3. (In Russ.)
18. Khayrullin R. Z., Application of the Bayesian approach in the problems of constructing statistical estimates when processing the results of testing measuring equipment, Vestnik metrologa, 2020, no. 1, pp. 9–15. (In Russ.)
19. Khayrullin R. Z., Volchkov A. A., Isaev Yu. A., Leonova K. S., Fufaeva O. V., Application of the Bayesian approach to construct eff ective estimates of measurement accuracy, Vestnik metrologa, 2020, no. 3, pp. 9–12. (In Russ.)
20. Khayrullin R. Z., Zakutin A. A., Measurement Techniques, 2021, vol. 63 no. 11, pp. 862–869. https://doi.org/10.1007/s11018-021-01872-x
21. Khayrullin R. Z., Khaimuldinova A. K., Taimanova G. K., Sarsembayeva T. E., Volkov V. S., Shamina S. V., Lyalin E. A., Kabanov O. V., Nexo Revista Cientifi ca, 2021, vol. 34, no. 4, pp. 1301– 1321. https://doi.org/10.5377/nexo.v34i04.12666
22. Khayrullin R. Z., Constructing the confi dence intervals and regions for a multiple linear regression model using the Bayesian approach, Al’manakh of modern metrology, 2021, no. 4, pp. 301–321. (In Russ.)
23. Rozental O. M., Alexandrovskaya L. N., Kirillin A. V., Bayesian approach to improving the reliability of water quality control, Analytics and control, 2018, vol. 22, no. 3, pp. 334–340. (In Russ.) http://dx.doi.org/10.15826/analitika.2018.22.3.001
Review
For citations:
Khayrullin R.Z. Multi-step algorithm for constructing statistical estimates based on the Bayesian approach in measuring tasks. Izmeritel`naya Tekhnika. 2022;(4):23-29. (In Russ.) https://doi.org/10.32446/0368-1025it.2022-4-23-29
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