Probabilistic graphical models that encode an underlying Markov random field are fundamental building blocks of generative modeling to learn latent representations in modern multivariate data sets with complex dependency structures. Among these, the exponential family graphical models are especially popular, given their fairly well-understood statistical properties and computational scalability to high-dimensional data based on pseudo-likelihood methods. These models have been successfully applied in many fields, such as the Ising model in statistical physics and count graphical models in genomics. Another strand of models allows some nodes to be latent, so as to allow the marginal distribution of the observable nodes to depart from exponential family to capture more complex dependence. These approaches form the basis of generative models in artificial intelligence, such as the Boltzmann machines and their restricted versions. A fundamental barrier to likelihood-based (i.e., both maximum likelihood and fully Bayesian) inference in both fully and partially observed cases is the intractability of the likelihood. The usual workaround is via adopting pseudo-likelihood based approaches, following the pioneering work of Besag (1974). The goal of this paper is to demonstrate that full likelihood based analysis of these models is feasible in a computationally efficient manner. The chief innovation lies in using a technique of Geyer (1991) to estimate the intractable normalizing constant, as well as its gradient, for intractable graphical models. Extensive numerical results, supporting theory and comparisons with pseudo-likelihood based approaches demonstrate the applicability of the proposed method.

翻译：编码潜在马尔可夫随机场的概率图模型是现代多元数据集中学习复杂依赖结构潜表示的基础生成模型构建模块。其中，指数族图模型因其相对完善的统计特性和基于伪似然方法对高维数据的计算可扩展性而尤为流行。此类模型已成功应用于多个领域，例如统计物理学中的Ising模型和基因组学中的计数图模型。另一类模型允许部分节点为潜变量，使可观测节点的边缘分布偏离指数族以捕获更复杂的依赖关系。这些方法构成了人工智能中生成模型（如玻尔兹曼机及其受限版本）的基础。在完全与部分可观测两种情形下，基于似然（即极大似然与完全贝叶斯）推断的根本障碍在于似然函数的难解性。传统解决途径遵循Besag（1974）的开创性工作采用伪似然方法。本文旨在证明，基于完整似然的模型分析在计算上是可行且高效的。核心创新在于运用Geyer（1991）的技术对具有难解归一化常数的图模型估计该常数及其梯度。大量数值结果、理论支撑以及与伪似然方法的比较验证了所提出方法的适用性。