When observations in a random sample of $n$ are classified into $k$ categories with $n_i$ falling into a given class, the usual estimate of the corresponding probability is $n_i/n$. When a priori information about the distribution of the probabilities is available, more precise estimates can be derived from data in any one sample of $n$. 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 the $n_i/n$ and their estimated sampling variances. The computations are analogous to those that arise in regression theory with the bivariate normal frequency distribution when $Y = X + e$ and the expected value of $X$ for a given $Y$ is estimated from the regression of $X$ on $Y$. The parameters of the distribution of $X$ have to be estimated from the observed distribution of $Y$ and the sampling errors in the individual values of those observations
Estimation of the Probability that An Observation Will Fall into a Specified Class
Hendricks, WA. (1964). Estimation of the Probability that An Observation Will Fall into a Specified Class. Journal of the American Statistical Association, 59(305), 225-232.