Asymptotics for Bayesian histograms.

We consider the problem of consistently estimating an unknown probability density function on a bounded interval from a sample of n independent and identically distributed univariate random variables. Adopting a Bayesian nonparametric approach, as first approximation, a hierarchical prior whose weak support comprises all absolutely continuous distribution functions, is considered that selects piecewise constant densities. The prior measure is constructed by putting a prior on the number of equal length bins and a Dirichtlet distribution on the bin values. Covergence rate for the Hellinger loss of the Bayes' estimator is deduced from the posterior rate, which is studied for various densities generating the data. This rate is comparable up to a logarithmic factor to that of the frequentist histogram estimator. Smoothing the Bayesian histogram, we get a continuous, piecewise linear competitor which possesses a faster rate of convergence.