T
he condition of detailed balance
has long been used as a proxy for t
he more difficult-to-prove condition of total balance, w
hic
h along wit
h ergodicity is required to guarantee convergence of a Markov C
hain Monte Carlo (MCMC) simulation to t
he correct probability distribution. However, some simple-to-program update sc
hemes suc
h as t
he sequential and c
heckerboard Metropolis algorit
hms are known not to satisfy detailed balance for suc
h common systems as t
he 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.
