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Register a new userAsymptotic Analysis of the Random-Walk Metropolis Algorithm on Ridged Densities
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Added by
Alexandre Thiery
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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In this paper we study the asymptotic behavior of the Random-Walk Metropolis algorithm on probability densities with two different `scales, where most of the probability mass is distributed along certain key directions with the `orthogonal directions containing relatively less mass. Such class of probability measures arise in various applied contexts including Bayesian inverse problems where the posterior measure concentrates on a sub-manifold when the noise variance goes to zero. When the target measure concentrates on a linear sub-manifold, we derive analytically a diffusion limit for the Random-Walk Metropolis Markov chain as the scale parameter goes to zero. In contrast to the existing works on scaling limits, our limiting Stochastic Differential Equation does not in general have a constant diffusion coefficient. Our results show that in some cases, the usual practice of adapting the step-size to control the acceptance probability might be sub-optimal as the optimal acceptance probability is zero (in the limit).
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In this article we propose multiplication based random walk Metropolis Hastings (MH) algorithm on the real line. We call it the random dive MH (RDMH) algorithm. This algorithm, even if simple to apply, was not studied earlier in Markov chain Monte Carlo literature. The associated kernel is shown to have standard properties like irreducibility, aperiodicity and Harris recurrence under some mild assumptions. These ensure basic convergence (ergodicity) of the kernel. Further the kernel is shown to be geometric ergodic for a large class of target densities on $mathbb{R}$. This class even contains realistic target densities for which random walk or Langevin MH are not geometrically ergodic. Three simulation studies are given to demonstrate the mixing property and superiority of RDMH to standard MH algorithms on real line. A share-price return data is also analyzed and the results are compared with those available in the literature.
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