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Abstract

A Bayesian method of estimation of a success probability $p$ is considered in the case when two experiments are available: individual Bernoulli $(p)$ trials—the $p$-experiment—or products of $r$ individual Bernoulli $(p)$ trials—the $p^{r}$-experiment. This problem has its roots in reliability, where one can test either single components or a system of $r$ identical components. One of the problems considered is to find the degree ${\tilde r}$ of the $p^{\tilde r}$-experiment and the size ${\tilde m}$ of the $p$-experiment such that the Bayes estimator based on ${\tilde m}$ observations of the $p$-experiment and $N-{\tilde m}$ observations of the $p^{\tilde r}$-experiment minimizes the Bayes risk among all the Bayes estimators based on $m$ observations of the $p$-experiment and $N-m$ observations of the $p^{r}$-experiment. Another problem is to sequentially select some combination of these two experiments, i.e., to decide, using the additional information resulting from the observation at each stage, which experiment should be carried out at the next stage to achieve a lower posterior expected loss.