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2007.91: Optimal Scaling of Random Walk Metropolis algorithms with Discontinuous target densities

2007.91: P Neal, G Roberts and J Yuen (2007) Optimal Scaling of Random Walk Metropolis algorithms with Discontinuous target densities.

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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
Additional Information:

Submitted to Annals of Applied Probability

Uncontrolled Keywords:Random walk Metropolis, Markov chain Monte Carlo, optimal scaling
Subjects:MSC 2000 > 60 Probability theory and stochastic processes
MSC 2000 > 65 Numerical analysis
MIMS number:2007.91
Deposited By:Dr Peter Neal
Deposited On:29 May 2007

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