#### 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 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 X1,...,X

*with distributions P*

_{n}_{1},...,P

*, respectively, given that X*

_{n}_{1}= ···= X

*. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions.*

_{n}*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

*into a single distribution*

_{n}*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 P

_{1},...,P

*are Gaussian,*

_{n}*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.

**URL:** http://digitalcommons.calpoly.edu/rgp_rsr/80