Guaranteeing total balance in Metropolis algorithm Monte Carlo simulations

Abstract

The condition of detailed balance has long been used as a proxy for the more difficult-to-prove condition of total balance, which along with ergodicity is required to guarantee convergence of a Markov Chain Monte Carlo (MCMC) simulation to the correct probability distribution. However, some simple-to-program update schemes such as the sequential and checkerboard Metropolis algorithms are known not to satisfy detailed balance for such common systems as the Ising model.It has been an open question whether these update schemes satisfy the weaker condition of total balance. In this work, we show that under fairly broad conditions, a large class of update schemes for the Metropolis algorithm, including the sequential and checkerboard schemes, do indeed satisfy total balance for important distributions. We also show that detailed balance itself can be satisfied by straightforward modifications to these schemes.