5 research outputs found
On the convergence of the Metropolis algorithm with fixed-order updates for multivariate binary probability distributions
The Metropolis algorithm is arguably the most fundamental Markov chain Monte
Carlo (MCMC) method. But the algorithm is not guaranteed to converge to the
desired distribution in the case of multivariate binary distributions (e.g.,
Ising models or stochastic neural networks such as Boltzmann machines) if the
variables (sites or neurons) are updated in a fixed order, a setting commonly
used in practice. The reason is that the corresponding Markov chain may not be
irreducible. We propose a modified Metropolis transition operator that behaves
almost always identically to the standard Metropolis operator and prove that it
ensures irreducibility and convergence to the limiting distribution in the
multivariate binary case with fixed-order updates. The result provides an
explanation for the behaviour of Metropolis MCMC in that setting and closes a
long-standing theoretical gap. We experimentally studied the standard and
modified Metropolis operator for models were they actually behave differently.
If the standard algorithm also converges, the modified operator exhibits
similar (if not better) performance in terms of convergence speed