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Asymptotic Analysis of the Random-Walk Metropolis Algorithm on Ridged Densities

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 نشر من قبل Alexandre Thiery
 تاريخ النشر 2015
  مجال البحث الاحصاء الرياضي
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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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