Title

Conflations of Probability Distributions

Author Info

Theodore P. Hill, California Polytechnic State University - San Luis ObispoFollow

Recommended Citation

Published in Transactions of the American Mathematical Society, Volume 363, Issue 6, June 1, 2011, pages 3351-3372.

This article was first published in Transactions of the American Mathematical Society, published by the American Mathematical Society. Copyright © 2011 American Mathematical Society.

The definitive version is available at https://doi.org/10.1090/S0002-9947-2011-05340-7.

Abstract

The conflation of a finite number of probability distributions P₁,...,P_n is a consolidation of those distributions into a single probability distribution Q = Q(P₁,...,P_n), where intuitively Q is the conditional distribution of independent random variables X1,...,X_n with distributions P₁,...,P_n, respectively, given that X₁ = ···= X_n. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the unique probability distribution that minimizes the loss of Shannon information in consolidating the combined information from P₁,...,P_n into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. In that sense, conflation may be viewed as an optimal method for combining the results from several different independent experiments. When P₁,...,P_n are Gaussian, Q is Gaussian with mean the classical weighted-mean-squares reciprocal of variances. A version of the classical convolution theorem holds for conflations of a large class of a.c. measures.