  The conflation of a finite number of probability distributions P_1,..., P_n
is a consolidation of those distributions into a single probability
distribution Q=Q(P_1,..., P_n), where intuitively Q is the conditional
distribution of independent random variables X_1,..., X_n with distributions
P_1,..., P_n, respectively, given that X_1= ... =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,..., 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. When
P_1,..., 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.
