When observations in a random sample ofnare classified intokcategories withnifalling into a given class, the usual estimate of the corresponding probability isni/n. When a priori information about the distribution of the probabilities is available, more precise estimates can be derived from data in any one sample ofn. When the a priori distribution is not specified completely, but its general form can be inferred, the parameters of that distribution can be estimated from the average of theni/nand their estimated sampling variances. The computations are analogous to those that arise in regression theory with the bivariate normal frequency distribution whenY=X+eand the expected value ofXfor a givenYis estimated from the regression ofXonY. The parameters of the distribution ofXhave to be estimated from the observed distribution ofYand the sampling errors in the individual values of those observations.