Optimal Scaling of Random Walk Metropolis algorithms with Discontinuous target densities

Neal, P and Roberts, G and Yuen, J (2007) Optimal Scaling of Random Walk Metropolis algorithms with Discontinuous target densities. [MIMS Preprint]

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Abstract

We consider the optimal scaling problem for high-dimensional Random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. 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 $\infty$. 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 optimising the efficiency of the RWM algorithm is equivalent to maximising 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.

Item Type: MIMS Preprint Submitted to Annals of Applied Probability Random walk Metropolis, Markov chain Monte Carlo, optimal scaling MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processesMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Dr Peter Neal 29 May 2007 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/811

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