SUMMARY This paper describes a Bayesian procedure for the simultaneous estimation of the proba- bilities in a histogram. A two-stage prior distribution is constructed which assumes that probabilities corresponding to adjacent intervals are likely to be closely related. The method employs multivariate logit transformations, and a covariance structure similar to that assumed in the first-order autoregressive process. Posterior estimates are obtained which combine information between the intervals and have the practical effect of smoothing the histogram. A weakness of the Bayesian approach has been its inability to cope with independent observations whose common distribution is not restricted to any particular family. We seek to remedy this deficiency by providing a technique for the analysis of n observations, which are assumed independent and identically distributed with unknown density q(y) which is concentrated on a finite interval I of the real line. We will assume that q(y) is thought a priori to possess a continuous first derivative for all y eI, or to possess some similar property of smoothness. We are posed with the problem of how to obtain estimates for q(y) and its moments which take account of this prior informa- tion. The problem will be treated by using a histogram to approximate q(y), and by esti- mating the probabilities in the histogram under the assumption that they are related in a certain manner. A disadvantage of our method is that the histogram estimate for q(y) will be discontinuous at several points in 1, and will not usually satisfy the smoothness property assumed a priori for the theoretical density. However, we hope that this will to some extent be compensated for by the advantages of the estimation procedure proposed for the probabilities in the histogram. Good & Gaskins (1971) and Boneva, Kendall & Stefanov (1971) provided sampling theory methods for the estimation of a density. An advantage of a Bayesian approach is that it takes proper account of the prior information, since the latter may be incorrectly emphasized when basing the analysis on intuitive ideas. Our method will be fairly flexible in allowing information about the degree of smoothness, and the shape, of the density to be incorporated into the prior model.