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Conflations of probability distributions


Author: Theodore P. Hill
Journal: Trans. Amer. Math. Soc. 363 (2011), 3351-3372
MSC (2000): Primary 60E05; Secondary 62B10, 94A17
Published electronically: January 5, 2011
MathSciNet review: 2775811
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Abstract: The conflation of a finite number of probability distributions $ P_1,\dots, P_n$ is a consolidation of those distributions into a single probability distribution $ Q=Q(P_1,\dots, P_n)$, where intuitively $ Q$ is the conditional distribution of independent random variables $ X_1,\dots, X_n$ with distributions $ P_1,\dots, P_n$, respectively, given that $ X_1=\cdots =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,\dots, 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. In that sense, conflation may be viewed as an optimal method for combining the results from several different independent experiments. When $ P_1,\dots, 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.


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Additional Information

Theodore P. Hill
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: hill@math.gatech.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05340-7
Keywords: Conflation of probability distributions, Shannon information, minimax likelihood ratio, proportional consolidation, product of probability density functions, product of probability mass functions, convolution theorem, Gauss-Markov theorem, best linear unbiased estimator, maximum likelihood estimator.
Received by editor(s): May 22, 2009
Received by editor(s) in revised form: February 26, 2010
Published electronically: January 5, 2011
Additional Notes: This work was partially supported by the Netherlands Organization for Scientific Research (NWO)
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.