Toward black-box sampling: A random-direction interior-point Markov chain approach

We briefly discuss issues concerning "black-box" algorithms for estimating properties of a k-dimensional posterior distribution with density pi(<(theta)under bar>\data). Typically, pi(<(theta)under bar>\data) is not known in closed form, but can be computed at any point <(theta)under bar> is an element of R(k) 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 <(theta)under bar>. We introduce an auxiliary variable Delta and consider a random variable (<(Theta)under bar>, Delta) from the interior of S, the region under the surface delta = pi(<(theta)under bar>\data) and above the plane delta = 0. Instead of directly sampling <(Theta)under bar> from pi(<(Theta)under bar>\data), RDIP generates a Markov chain {(<(Theta)under bar>(i), Delta(i)), i > 0} from the uniform distribution on S. Then the Markov chain {<(Theta)under bar>(i), i greater than or equal to 0} has a unique stationary distribution pi(<(Theta)under bar>\data). 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.