Spatial Monte Carlo integration (SMCI) is an extension of standard Monte Carlo integration and can approximate expectations on Markov random fields with high accuracy. SMCI was applied to pairwise Boltzmann machine (PBM) learning, achieving superior results over those of some existing methods. The approximation level of SMCI can be altered, and it was proved that a higher-order approximation of SMCI is statistically more accurate than a lower-order approximation. However, SMCI as proposed in previous studies suffers from a limitation that prevents the application of a higher-order method to dense systems. This study makes two contributions. First, a generalization of SMCI (called generalized SMCI (GSMCI)) is proposed, which allows a relaxation of the above-mentioned limitation; moreover, a statistical accuracy bound of GSMCI is proved. Second, a new PBM learning method based on SMCI is proposed, which is obtained by combining SMCI and persistent contrastive divergence. The proposed learning method significantly improves learning accuracy.

空间蒙特卡罗积分 (SMCI) 是标准蒙特卡罗积分的扩展，可以高精度地近似马尔可夫随机场的预期。SMCI 被应用于成对玻尔兹曼机 (PBM) 学习，取得了优于一些现有方法的结果。SMCI 的近似水平可以改变，并且证明了 SMCI 的高阶近似在统计上比低阶近似更准确。然而，先前研究中提出的 SMCI 受到限制，阻止高阶方法应用于密集系统。这项研究有两个贡献。首先，提出了SMCI的泛化（称为广义SMCI（GSMCI）），它可以放宽上述限制；此外，证明了GSMCI的统计精度界限。第二，提出了一种新的基于SMCI的PBM学习方法，该方法是结合SMCI和持久对比发散获得的。所提出的学习方法显着提高了学习准确性。