Non-linear Markov Chain Monte Carlo

¹ Department of Mathematics, University of Bristol, UK
² Department of Mathematics, Imperial College London, UK
³ Departments of Statistics & Computer Science, University of British Columbia, CA
⁴ Department of Mathematics, University of Nice Sophia Antipolis, FR

Abstract

In this paper we introduce a class of non-linear Markov Chain Monte Carlo (MCMC) methods for simulating from a probability measure π. Non-linear Markov kernels (e.g. Del Moral (2004)) can be constructed to admit π as an invariant distribution and have typically superior mixing properties to ordinary (linear) MCMC kernels. However, such non-linear kernels often cannot be simulated exactly, so, in the spirit of particle approximations of Feynman-Kac formulae (Del Moral 2004), we construct approximations of the non-linear kernels via Self-Interacting Markov Chains (Del Moral & Miclo, Proc. R. Soc. Lond. A, 460, 325-46, 2004.) (SIMC). We present several non-linear kernels and investigate the performance of our approximations with some simulations.