Approximate Random Variate Generation from Infinitely Divisible Distributions with Applications to Bayesian Inference

Abstract

SUMMARY Stochastic processes with independent increments play a central role in Bayesian nonparametric inference. The distributions of the increments of these processes, aside from fixed points of discontinuity, are infinitely divisible and their Laplace and/or Fourier transforms in the Lévy representation are usually known. Conventional Bayesian inference in this context has been limited largely to providing point estimates of the random quantities of interest, although Markov chain Monte Carlo methods have been used to obtain a fuller analysis in the context of Dirichlet process priors. In this paper, we propose and implement a general method for simulating infinitely divisible random variates when their Fourier or Laplace transforms are available in the Lévy representation. Theoretical justification is established by proving a convergence theorem that is a ‘sampling form’ of a classical theorem in probability. The results provide a method for implementing Bayesian nonparametric inference by using a wide range of stochastic processes as priors.