Bayesian Inference for Multivariate Monotone Densities

ABSTRACT We consider a nonparametric Bayesian approach to estimation and testing for a multivariate monotone density. Instead of following the conventional Bayesian approach of imposing a prior that satisfies the monotonicity restriction, we place a prior on the step heights via binning and a Dirichlet distribution. The resulting posterior distribution is conjugate, but a step function with Dirichlet distributed heights may not comply with the monotonicity restriction. We transform an arbitrary piecewise constant probability density into a monotone one by its ‐projection onto the space of multivariate monotone functions, and subsequently normalize it to integrate to one. This map is then used to induce a “projection posterior distribution” on the space of multivariate monotone densities from the conjugate posterior by treating the unconstrained density as a fundamental parameter and the multivariate monotone density as a transformation of it, and to infer. We show that the resulting projection posterior contracts at the optimal rate. We also construct consistent Bayesian tests to test multivariate monotonicity of a probability density based on the ‐distance to the class of monotone functions. The test is shown to have a size approaching zero and high power against alternatives sufficiently separated from the null hypothesis. To obtain a Bayesian credible interval for the value of the density function at an interior point with guaranteed asymptotic frequentist coverage, we consider a posterior quantile interval of an induced map transforming the function value to its value optimized over certain blocks. The limiting coverage is explicitly calculated and is found to exceed the credibility level used in the construction. By examining the asymptotic relationship between coverage and credibility, we show that a desired asymptotic coverage can be achieved precisely by starting with an appropriate credibility level. We assess the accuracy of our approach compared to a monotonized conventional density estimator. Furthermore, we analyze a real‐world dataset on joint p ‐values for gene expressions.