Toward Black-Box Sampling: A Random-Direction Interior-Point Markov Chain Approach

Abstract

We briefly discuss issues concerning “black-box” algorithms for estimating properties of a κ-dimensional posterior distribution with density . Typically, is not known in closed form, but can be computed at any point as the product of the likelihood function and the prior density. Black-box algorithms provide estimates to a user who must specify only minimal information about the distribution, the properties to be estimated, and the desired precision of the estimates. Ideally, reasonable performance should require no tuning of algorithm parameters. We propose a random-direction interior-point (RDIP) Markov chain approach to black-box sampling. RDIP requires only that the product of the likelihood function and the prior density can be computed at any point . We introduce an auxiliary variable Δ and consider a random variable from the interior of S, the region under the surface and above the plane δ = 0. Instead of directly sampling from , RDIP generates a Markov chain from the uniform distribution on S. Then the Markov chain has a unique stationary distribution . We develop variations of the RDIP samplers, study their performance, and discuss algorithm parameter values. Three examples are presented, two illustrative examples that use RDIP alone, and a more complex application that takes RDIP steps within a Gibbs sampler.

Key Words:

• Bayesian computation
• Gibbs sampler
• Hit-and-Run
• Metropolis-Hastings algorithm
• Monte Carlo
• Multidimensional integrals
• Posterior distributions
• Prior distributions
• Simulation