Truncated Multivariate Student Computations via Exponential Tilting

In this paper we consider computations with the multivariate student density, truncated on a set described by a linear system of inequalities. Our goal is to both simulate from this truncated density, as well as to estimate its normalizing constant. To this end we consider an exponentially tilted sequential importance sampling (IS) density. We prove that the corresponding IS estimator of the normalizing constant, a rare-event probability, has bounded relative error under certain conditions. Along the way, we establish the multivariate extension of the Mill’s ratio for the student distribution. We present applications of the proposed sampling and estimation algorithms in Bayesian inference. In particular, we construct efficient rejection samplers for the posterior densities of the Bayesian Constrained Linear Regression model, the Bayesian Tobit model, and the Bayesian smoothing spline for non-negative functions. Typically, sampling from such posterior densities is only viable via approximate Markov chain Monte Carlo (MCMC). Finally, we propose a novel Reject-Regenerate sampler, which is a hybrid between rejection sampling and MCMC. The Reject-Regenerate sampler creates a Markov chain, whose states are, with a certain probability, flagged as commencing a new regenerative or renewal cycle. Whenever a state initiates a new regenerative cycle, we can further flip a biased coin to decide whether the state is an exact draw from the target, or not. We show that the proposed MCMC algorithm is strongly efficient in a rare-event regime and provide a numerical example.