Approximate Random Variate Generation From Infinitely Divisible Distributions With Applications to Bayesian Inference

Authors: 
P. Damien, P.W. Laud, A.F.M. Smith
Imperial College, Kimberly-Clark Corp., Medical College of Wisconsin

Nov 30 1993

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 int he Levy representation are usually known. Conventional Bayesian literature 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/Laplace transforms are available in the Levy 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 implementic Bayesian nonparametric inference using a wide range of stochastic processes as priors.

Keywords: 

infinitely divisible distributions, Levy process, Independent increments, characteristic functions, moment generating functions, stable laws, fourier transforms, laplace transforms, importance functions, bayesian nonparametric inference

Manuscript: 

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