Optimal scaling of random walk Metropolis algorithms with discontinuous target densities

We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to ∞. In particular, when the proposal variance is scaled by d−2, the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of e−2 (=0.1353) under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.

[1]  J. Doob.  Stochastic processes , 1953 .

[2]  A. Barbour.  Stein's method and poisson process convergence , 1988, Journal of Applied Probability.

[3]  李幼升,et al.  Ph , 1989 .

[4]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[5]  A. Barbour,et al.  Poisson Approximation , 1992 .

[6]  P. Spreij.  Probability and Measure , 1996 .

[7]  A. Gelman,et al.  Weak convergence and optimal scaling of random walk Metropolis algorithms , 1997 .

[8]  J. Rosenthal,et al.  Optimal scaling of discrete approximations to Langevin diffusions , 1998 .

[9]  From Metropolis to Diffusions: Gibbs States and Optimal Scaling , 1998 .

[10]  Sergey Utev,et al.  Branching Approximation for the Collective Epidemic Model , 1999 .

[11]  J. Rosenthal,et al.  Optimal scaling for various Metropolis-Hastings algorithms , 2001 .

[12]  Frank Ball,et al.  Stochastic multi-type SIR epidemics among a population partitioned into households , 2001, Advances in Applied Probability.

[13]  Sergio Scarlatti,et al.  Optimal scaling of MaLa for nonlinear regression , 2004 .

[14]  J. Rosenthal,et al.  Scaling limits for the transient phase of local Metropolis–Hastings algorithms , 2005 .

[15]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[16]  C. Sherlock.  Methodology for inference on the Markov modulated Poisson process and theory for optimal scaling of the random walk Metropolis , 2006 .

[17]  G. Roberts,et al.  OPTIMAL SCALING FOR PARTIALLY UPDATING MCMC ALGORITHMS , 2006, math/0607054.

[18]  M. B'edard.  Weak convergence of Metropolis algorithms for non-i.i.d. target distributions , 2007, 0710.3684.

[19]  M. Bédard.  Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234 , 2008 .

[20]  G. Roberts,et al.  Optimal Scaling for Random Walk Metropolis on Spherically Constrained Target Densities , 2008 .

[21]  G. Roberts,et al.  Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets , 2009, 0909.0856.

[22]  G. Roberts,et al.  Optimal Scaling of Random Walk Metropolis Algorithms with Non-Gaussian Proposals , 2011 .

[23]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .