#### 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 *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.

#### Disciplines

Mathematics

**URL:** http://digitalcommons.calpoly.edu/rgp_rsr/17