Posterior asymptotics in the supremum $L_{1}$ norm for conditional density estimation

Abstract

In this paper we study posterior asymptotics for conditional density estimation in the supremum $L_{1}$ norm. Compared to the expected $L_{1}$ norm, the supremum $L_{1}$ norm allows accurate prediction at any designated conditional density. We model the conditional density as a regression tree by defining a data dependent sequence of increasingly finer partitions of the predictor space and by specifying the conditional density to be the same across all predictor values in a partition set. Each conditional density is modeled independently so that the prior specifies a type of dependence between conditional densities which disappears after a certain number of observations have been observed. The rate at which the number of partition sets increases with the sample size determines when the dependence between pairs of conditional densities is set to zero and, ultimately, drives posterior convergence at the true data distribution.