Conflations of probability distributions

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.