A remark on the converse of Laplace's theorem

Abstract

UDC 519.21 1. At the beginning of the 1940's S. N. Bernshtein established that in the case of the binomial distribution the a posteriori distribution of the probability of positive outcome as the number of experiments increases without bound is practically independent of the a priori distribution. Bernshtein proved this fact under the assumption that the binomial distribution satisfies the hypotheses of the central limit theorem (in the present case, the hypotheses of Laplace's theorem). In this situation it was shown that the a posteriori distribution function of the probability of a positive outcome can be approximated by the normal approximation to the binomial distribution. Bernshtein called this theorem the converse of Laplace's theorem [1]. These facts were later included in an appendix to Bernshtein's famous textbook Theory of Probability, where in addition it was proved that when a consistent estimate exists for the random shift parameter, the corresponding a posteriori distribution is practically independent of the a priori distribution in a large number of cases and can be approximated by the conditional distribution of the consistent estimate. Thus the latter theorem is stronger than the converse of Laplace's theorem, which relies on an additional requirement (namely that the hypotheses of the central limit theorem be satisfied). Moreover it should be noted that both of the Bernshtein theorems mentioned are proved under the assumption that the value of the a priori density at some random point is strictly positive. Therefore, strictly speaking, these theorems are not absolutely valid, but only with a certain probability. The purpose of the present article is to show that in the case of the binomial distribution the a posteriori distribution of the probability of a positive outcome can be approximated (in the sense of the strong law of large numbers) by a binomial distribution close to the original one and is practically independent of the a priori distribution (which Bernshtein had assumed continuous). This fact is unconnected with the theorem of Laplace and makes it possible to find an approximation for the a posteriori distribution even in the case when the binomial distribution is close to a Poisson distribution. We note here that it was L. N. Bol'shev who posed the actual problem.