Change of Measure in Monte Carlo Integration via Gibbs Sampling with an application to Stochastic Volatility Models

In this article we focus on how to combine important sampling techniques within Markov chain Monte Carlo methods, particularly the Gibbs sampler. We propose a way for constructing a change of measure with respect to the intractable target joint distribution of the sampler by\ employing proper augmenting functions of the conditional distributions. The almost certain convergence of the estimates under the new measure will be proved, and it will be shown that, under the new measure, the variance of the Monte Carlo integration can be reduced. More interestingly, the change of measure induces a modification of the Markov chain generated by the sampler and, for a proper choice of the important sampling function, the new chain will show better mixing properties than the original. This is all illustrated in an application to the smoothing of the underlying conditional variance in a stochastic volatility model.