A nonparametric process for multi-resolution adaptive shrinkage

We introduce a nonparametric prior for probability densities that achieves multi-resolution adaptive shrinkage in posterior inference. The prior applies a varying amount of shrinkage to data structures of different scales and/or at different locations. It is adaptive in that the appropriate amount of shrinkage is determined through the behavior of the data at different scale-location combinations. The prior employs two important additional features: a Markov feature that introduces dependence into the shrinkage across different scale-location combinations, and a randomized partitioning feature that allows adaptive construction of multi-scale grids to most effectively characterize the underlying distribution. We study the theoretical properties of the process, showing that it possesses large support, posterior conjugacy, and posterior consistency. We then provide analytic recipes for marginalization and for computing the posterior through recursion. We illustrate through several numerical examples that the multi-resolution adaptiveness so achieved can substantially improve inference in nonparametric problems.