Efficient adaptive MCMC through precision estimation

TL;DR

This paper introduces an efficient adaptive MCMC algorithm that leverages sparsity in the precision matrix to improve covariance estimation and sampling efficiency for complex densities.

Contribution

It presents a novel method combining partial correlation structure estimation with precision matrix estimation for adaptive MCMC, applicable to general densities with unknown dependencies.

Findings

Faster convergence of covariance estimates compared to empirical methods

Lower computational cost per iteration

Effective for general densities with unknown dependency structures

Abstract

A novel adaptive Markov chain Monte Carlo algorithm is presented. The algorithm utilizes sparsity in the partial correlation structure of a density to efficiently estimate the covariance matrix through the Cholesky factor of the precision matrix. The algorithm also utilizes the sparsity to sample efficiently from both MALA and Metropolis Hasting random walk proposals. Further, an algorithm that estimates the partial correlation structure of a density is proposed. Combining this with the Cholesky factor estimation algorithm results in an efficient black-box AMCMC method that can be used for general densities with unknown dependency structure. The method is compared with regular empirical covariance adaption for two examples. In both examples, the proposed method's covariance estimates converge faster to the true covariance matrix and the computational cost for each iteration is lower.