Title

Conflations of Probability Distributions

Author Info

Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow

Recommended Citation

Available from Mathematics arXiv, August 1, 2008.

23 pages.

Copyright © 2008 by Theodore P. Hill. The definitive version is available at http://arxiv.org/abs/0808.1808.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

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. 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.

Disciplines

Mathematics