Los Alamos Report LA-UR-02-4105, summary of talk presented at *7th Valencia International Meeting on Bayesian Statistics*, June 5, 2002, Tenerife, Spain

## Use of Probability Gradients in Hybrid MCMC and a New Convergence Test

Kenneth M. Hanson

*Los Alamos National Laboratory*
### Abstract

The hybrid Markov Chain Monte Carlo technique affords a robust means for
sampling multidimensional probability density functions with high
efficiency, provided one can calculate the gradient of phi =
minus-log-probability in a time comparable to calculating the probability
itself. The latter condition is met using the technique of adjoint
differentiation, which is explained. The gradient of phi may also be used to form a measure of the degree of convergence of the MCMC sequence. The proposed statistic is the ratio of sample estimates for the variance of the distribution calculated two different ways, one based on using integration by parts of the integral for the variance and the other based on the normal second-moment calculation. The efficiency of the hybrid MCMC technique and the usefulness of the convergence test will be demonstrated for simple multivariate normal distributions.

adjoint differentiation; Markov Chain Monte Carlo; hybrid MCMC; Hamiltonian method; Hamiltonian dynamics; leapfrog method; Metropolis MCMC; convergence test; multivariate distributions; simulation science; simulation validation
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Send e-mail to author at kmh@hansonhub.com