Bayesian shape-constrained density estimation
Sutanoy Dasgupta, Debdeep Pati, Anuj Srivastava · Quarterly of Applied Mathematics · 2019
The problem of estimating probability densities underlying given i.i.d. samples is a fundamental problem in statistics. Taking a Bayesian nonparametric approach, we put forth a geometric solution that uses different actions of the diffeomorphism (domain warping) group on the set of positive pdf s to explore this space more efficiently. This representation shifts the focus from pdf s to the diffeomorphism group and allows efficient solutions for density estimation under shape (or modality) constraints, i.e., estimation of a pdf given a fixed or a maximum number of modes. Focusing on univariate density estimation, we use the geometry of a (one-dimensional) diffeomorphism group to reach an (approximate) finite-dimensional Euclidean representation of warping functions, and impose a shrinkage prior on this space to form a posterior distribution. We sample this posterior using the Markov Chain Monte Carlo algorithm and form Bayesian estimates of the unknown pdf . This framework results in a novel pdf estimator, with and without shape constraints, and we demonstrate it in a number of simulated and real data experiments.