Variational method for estimating the rate of convergence of Markov Chain Monte Carlo algorithms

Details Variational method for estimating the rate of convergence of Markov Chain Monte Carlo algorithms Variational method for estimating the rate of convergence of Markov Chain Monte Carlo algorithms

2006-08-31

arxiv.org/abs/physics/0609001v3

Physics

Data Analysis, Statistics and Probability

27 pages, 8 figures. Added paragraph to intro and results section, format for Phys. Rev. E

Scientific paper

We demonstrate the use of a variational method to determine a quantitative lower bound on the rate of convergence of Markov Chain Monte Carlo (MCMC) algorithms as a function of the target density and proposal density. The bound relies on approximating the second largest eigenvalue in the spectrum of the MCMC operator using a variational principle and the approach is applicable to problems with continuous state spaces. We apply the method to one dimensional examples with Gaussian and quartic target densities, and we contrast the performance of the Random Walk Metropolis-Hastings (RWMH) algorithm with a ``smart'' variant that incorporates gradient information into the trial moves. We find that the variational method agrees quite closely with numerical simulations. We also see that the smart MCMC algorithm often fails to converge geometrically in the tails of the target density except in the simplest case we examine, and even then care must be taken to choose the appropriate scaling of the deterministic and random parts of the proposed moves. Again, this calls into question the utility of smart MCMC in more complex problems. Finally, we apply the same method to approximate the rate of convergence in multidimensional Gaussian problems with and without importance sampling. There we demonstrate the necessity of importance sampling for target densities which depend on variables with a wide range of scales.

Affiliated with

Also associated with

No associations

Rating

Variational method for estimating the rate of convergence of Markov Chain Monte Carlo algorithms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Variational method for estimating the rate of convergence of Markov Chain Monte Carlo algorithms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Variational method for estimating the rate of convergence of Markov Chain Monte Carlo algorithms will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-492494