Composite Spatial Monte Carlo Integration Based on Generalized Least Squares

Although evaluation of the expectations on the Ising model is essential in various applications, this is frequently infeasible because of intractable multiple summations (or integrations). Spatial Monte Carlo integration (SMCI) is a sampling-based approximation, and can provide high-accuracy estimations for such intractable expectations. To evaluate the expectation of a function of variables in a specific region (called target region), SMCI considers a larger region containing the target region (called sum region). In SMCI, the multiple summation for the variables in the sum region is precisely executed, and that in the outer region is evaluated by the sampling approximation such as the standard Monte Carlo integration. It is guaranteed that the accuracy of the SMCI estimator is monotonically improved as the size of the sum region increases. However, a haphazard expansion of the sum region could cause a combinatorial explosion. Therefore, we hope to improve the accuracy without such region expansion. In this study, based on the theory of generalized least squares, a new effective method is proposed by combining multiple SMCI estimators. The validity of the proposed method is demonstrated theoretically and numerically. The results indicate that the proposed method can be effective in the inverse Ising problem (or Boltzmann machine learning).