Conflations of Probability Distributions

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. AMS 2000 Classification: Primary: 60E05; Secondary: 62B10, 94A17