Conflations of probability distributions

Abstract

The conflation of a finite number of probability distributions P 1 , … , P n P_1,\dots , P_n is a consolidation of those distributions into a single probability distribution Q = Q ( P 1 , … , P n ) Q=Q(P_1,\dots , P_n) , where intuitively Q Q is the conditional distribution of independent random variables X 1 , … , X n X_1,\dots , X_n with distributions P 1 , … , P n P_1,\dots , P_n , respectively, given that X 1 = ⋯ = X n 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 Q is shown to be the unique probability distribution that minimizes the loss of Shannon information in consolidating the combined information from P 1 , … , P n P_1,\dots , P_n into a single distribution Q 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 , … , P n P_1,\dots , P_n are Gaussian, Q 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.