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.



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