Concentration rate and consistency of the posterior under monotonicity constraints

In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non increasing densities in a Bayesian setting and derive concentration rate for the posterior distribution for a Dirichlet process and finite mixture prior. We prove that the posterior distribution based on both priors concentrates at the rate $(n\mathrm{/}\log (n){)}^{-1\mathrm{/}3}$, which is the minimax rate of estimation up to a \log(n)$ factor. We also study the behaviour of the posterior for the point-wise loss at any fixed point of the support the density and for the sup norm. We prove that the posterior is consistent for both losses.