Conflations of Probability Distributions

TL;DR

This paper introduces the concept of conflation as a method to combine multiple probability distributions into a single optimal distribution, minimizing information loss and satisfying certain likelihood criteria, with specific results for Gaussian cases.

Contribution

It formalizes conflation as a unique optimal distribution for consolidating multiple distributions, extending classical convolution results to a broader class of measures.

Findings

Conflation minimizes Shannon Information loss when combining distributions.

For Gaussian distributions, conflation results in a weighted mean of means with reciprocal variances.

A generalized convolution theorem applies to conflations of absolutely continuous measures.

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.