Mean Field Initialization of the Annealed Importance Sampling Algorithm for an Efficient Evaluation of the Partition Function Using Restricted Boltzmann Machines

Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the partition function of magnetic spin and spin-like systems mapped into RBMs using AIS. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of Z following an annealing process. We show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost. The procedures presented are not linked to any learning process, and therefore do not require a priori knowledge of a training dataset.