Conflations of Probability Distributions
Abstract
The conflation of a finite number of probability distributions P_1,..., P_n is a consolidation of those distributions into a single probability distribution Q=Q(P_1,..., P_n), where intuitively Q is the conditional distribution of independent random variables X_1,..., X_n with distributions P_1,..., P_n, respectively, given that X_1= ... =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 P_1,..., 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 P_1,..., 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.
Cite
@article{arxiv.0808.1808,
title = {Conflations of Probability Distributions},
author = {Theodore P. Hill},
journal= {arXiv preprint arXiv:0808.1808},
year = {2009}
}
Comments
Additional reference, revised abstract, revised introduction (idempotency), revised title, and revised introduction to section 7; these changes plus converting the manuscript from plain tex to latex shortened the paper to 23 pages