Multi-Step Algorithm for Constructing Statistical Estimates Based on the Bayesian Approach in Measurement Problems

A multi-step algorithm based on the Bayesian approach to construct effective statistical estimates of the measurement results has been developed. A general logical scheme of a multi-step algorithm is presented, which in the case of heterogeneity of a priori and a posteriori data, enables the construction of generalized Bayesian estimates based on a mixture of a priori and a posteriori distributions. Conditions for the existence of a conjugate family of a priori distribution are formulated. A technique for calculating the specific parameter values in conjugation with a priori distribution is described. The technology for applying the Bayesian approach is described in detail for the binomial and negative binomial distribution laws. Using the likelihood function, the equations for recalculating the parameters of the corresponding conjugate distribution law are obtained, required for stepwise transitions in the multi-step algorithm. Examples of applying the algorithm for assessing the compliance of measuring systems with specified requirements, as well as for processing the results of quantitative physical and chemical analyses are presented. The results demonstrate that the Bayesian approach significantly exceeds the maximum likelihood method in terms of the accuracy of constructing statistical estimates for small and medium sample sizes. This circumstance makes the Bayesian approach particularly effective in assessing the metrological characteristics of measuring systems when multiple test repetitions are inexpedient or time consuming. Some examples illustrate that with the increasing volume and number of samples, the multi-step Bayesian approach and the classical maximum likelihood method will give identical results.