APPROXIMATE RANDOM VARIATE GENERATION FROM INFINITELY DIVISIBLE DISTRIBUTIONS WITH APPLICATIONS TO BAYESIAN-INFERENCE

Stochastic processes with independent increments play a central role i n Bayesian nonparametric inference. The distributions of the increment s of these processes, aside from fixed points of discontinuity, are in finitely divisible and their Laplace and/or Fourier transforms in the Levy 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 Car lo methods have been used to obtain a fuller analysis in the context o f Dirichlet process priors. In this paper, we propose and implement a general method for simulating infinitely divisible random variates whe n their Fourier or Laplace transforms are available in the Levy repres entation. Theoretical justification is established by proving a conver gence theorem that is a 'sampling form' of a classical theorem in prob ability. The results provide a method for implementing Bayesian nonpar ametric inference by using a wide range of stochastic processes as pri ors.