Skip to main content
Article
Conflations of Probability Distributions
Transactions of the American Mathematical Society (2011)
  • Theodore P. Hill, California Polytechnic State University - San Luis Obispo
Abstract

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. In that sense, conflation may be viewed as an optimal method for combining the results from several different independent experiments. 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.

Disciplines
Publication Date
June, 2011
Publisher Statement
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 http://dx.doi.org/10.1090/S0002-9947-2011-05340-7.
Citation Information
Theodore P. Hill. "Conflations of Probability Distributions" Transactions of the American Mathematical Society Vol. 363 Iss. 6 (2011)
Available at: http://works.bepress.com/tphill/79/