Model selection of stochastic simulation algorithm based on generalized divergence measures

Abstract

MCMC methods (Monte Carlo Markov Chain) are a class of methods used to perform simulations per a probability distribution $P$. These methods are often used when we have difficulties to directly sample per a given probability distribution $P$ . This distribution is then considered as a target and generates a Markov chain $(X_n)_{n\in\mathbb{N}}$ that, when $n$ is large we have $X_n\sim P$. These MCMC methods consist of several simulation strategies including the \emph{Independent Sampler (IS)}, the \emph{Random Walk of Metropolis Hastings \small{(RWMH)}}, the \emph{Gibbs sampler}, the \emph{Adaptive Metropolis (AM)} and \emph{Metropolis Within Gibbs (MWG)} strategy. Each of these strategies can generate a Markov chain and is associated with a convergence speed. It is interesting, with a given target law, to compare several simulation strategies for determining the best. Chauveau and Vandekerkhove \cite{Chauv2007} have compared IS and RWMH strategies using the Kullback-Leibler divergence measure. In our article we will compare our five simulation methods already mentioned using generalized divergence measures. These divergence measures are taken in family of $\alpha$-divergence measures \cite{Cichocki2010}, with a parameter $\alpha$. This is the R\'enyi divergence, Tsallis divergence and $D_\alpha$ divergence .