Diffusive limits for Adaptive MCMC

Details Diffusive limits for Adaptive MCMC Diffusive limits for Adaptive MCMC

2012-01-06

arxiv.org/abs/1201.1433v1

Mathematics

Probability

It has 21 pages including 6 tables. One of the authors presented part of it in a seminar at the department of Mathematics, Uni

Scientific paper

Adaptive Markov Chain Monte Carlo (AMCMC) are a class of algorithms that have been recently proposed in MCMC literature. The main difference between MCMC and AMCMC is that, the tuning parameter, which determines how fast the simulation converges to the desired distribution {\psi}(\bullet), is a function of the previous sample paths. This destroys the Markovian character of the chain. However it can be shown that, under some conditions, the adaptive chain converges to the target distribution {\psi}. In this paper we use a diffusion approximation technique on a discrete time AMCMC. The resulting diffusion, which is a two-dimensional degenerate one, gives some idea of the dynamics of the chain. Next by extensive simulation, for various target distributions (both heavy and light tailed, symmetric and asymmetric), we show that the marginal of the invariant distribution is, indeed, the target distribution. Using simulation methods the speed of the convergence is also compared with that of random-walk Metropolis Hastings sampler.

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