  Although evaluation of the expectations on the Ising model is essential in
various applications, it is mostly infeasible because of intractable multiple
summations. Spatial Monte Carlo integration (SMCI) is a sampling-based
approximation. It 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 improves monotonically 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 an
expansion. In this paper, based on the theory of generalized least squares
(GLS), 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).
