A Posteriori Integration of Probabilities. Elementary Theory

Kryazhimskiy, A. (2016). A Posteriori Integration of Probabilities. Elementary Theory. Theory of Probability & Its Applications 60 (1) 62-87. 10.1137/S0040585X97T987466.

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Abstract

An approach to a posteriori integration of probability distributions serving as independent a priori models of observed elementary events from a given finite set of elementary events is proposed. A posteriori integration is understood as an improvement of data given by a priori probabilities. The approach is based on the concept of an a posteriori event in the product of probability spaces associated with a priori probabilities. The conditional probability on the product space that is specified by an a posteriori event determines in a natural way the probability on the set of initial elementary events; the latter is recognized as the result of a posteriori integration of a priori models. Conditions under which the integration improves the informativeness of a priori probabilities are established, algebraic properties of integration as a binary operation on the set of probabilities are studied, and the problem of integral convergence of infinite probability sequences is considered.

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