Spatial Monte Carlo Integration with Annealed Importance Sampling

Evaluating expectations on a pairwise Boltzmann machine (PBM) (or Ising model) is important for various applications, including the statistical machine learning. However, in general the evaluation is computationally difficult because it involves intractable multiple summations or integrations; therefore, it requires an approximation. Monte Carlo integration (MCI) is a well-known approximation method; a more effective MCI-like approximation method was proposed recently, called spatial Monte Carlo integration (SMCI). However, the estimations obtained from SMCI (and MCI) tend to perform poorly in PBMs with low temperature owing to degradation of the sampling quality. Annealed importance sampling (AIS) is a type of importance sampling based on Markov chain Monte Carlo methods, and it can suppress performance degradation in low temperature regions by the force of importance weights. In this study, a new method is proposed to evaluate the expectations on PBMs combining AIS and SMCI. The proposed method performs efficiently in both high- and low-temperature regions, which is theoretically and numerically demonstrated.