Conflations of Probability Distributions
Abstract
Description
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.
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
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