Metropolis walking sampling

The step sizes for a typical Metropolis walk are usually chosen to give an acceptance ratio of about one-half in order to maximize the rate of diffusion and improve the sampling speed. Serial correlation of points is usually high. In many-dimensional (or many-electron) systems, the steps may be taken one dimension (or one electron) at a time or all at once. The optimum step sizes and/or combinations of steps depend strongly on the nature of the system treated. 

Another alternative, likely to be more efficient than Metropolis sampling, is the use of probability density functions P. These relatively simple functions, which approximate and mimic the density of the more complex function V / , can be sampled directly without a Metropolis walk and the associated serial correlation. Sample points of unit weight are obtained with probabilities proportional to the probability density P, and their weights are multiplied by the factor if /P to give overall v /o weighting. The expectation value of the energy ) is then given by... 

In the Monte Carlo method to estimate a many-dimensional integral by sampling the integrand. Metropolis Monte Carlo or, more generally, Markov chain Monte Carlo (MCMC), to which this volume is mainly devoted, is a sophisticated version of this where one uses properties of random walks to solve problems in high-dimensional spaces, particularly those arising in statistical mechanics. 

Specifically, this involves generating a Markov chain of steps by box sampling R = R + qA, with A the box size, and q a 3M-dimensional vector of uniformly distributed random numbers q e [— 1, +1]. This is followed by the classic Metropolis accept/reject step, in which ( PX(R )/T,X(R))2 is compared to a uniformly distributed random number between zero and unity. The new coordinate R is accepted only if this ratio of trial wavefunctions squared exceeds the random number. Otherwise the new coordinate remains at R. This completes one step of the Markov chain (or random walk). Under very general conditions, such a Markov chain results in an asymptotic equilibrium distribution proportional to, FX(R). Once established, the properties of interest can be measured at each point R in the Markov chain (which we refer to as a configuration) using Eqs. (1.2) and... 

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