Available from Mathematics arXiv, August 1, 2008.
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.
The conflation of a finite number of probability distributions P1,..., Pn is a consolidation of those distributions into a single probability distribution Q=Q(P1,..., Pn), where intuitively Q is the conditional distribution of independent random variables X1,..., Xn with distributions P1,..., Pn, respectively, given that X1= ... =Xn. 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 P1,..., Pn into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. When P1,..., Pn 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.