A Markov network characterizes the conditional independence structure, or Markov property, among a set of random variables. Existing work focuses on specific families of distributions (e.g., exponential families) and/or certain structures of graphs, and most of them can only handle variables of a single data type (continuous or discrete). In this work, we characterize the conditional independence structure in general distributions for all data types (i.e., continuous, discrete, and mixed-type) with a Generalized Precision Matrix (GPM). Besides, we also allow general functional relations among variables, thus giving rise to a Markov network structure learning algorithm in one of the most general settings. To deal with the computational challenge of the problem, especially for large graphs, we unify all cases under the same umbrella of a regularized score matching framework. We validate the theoretical results and demonstrate the scalability empirically in various settings.

翻译：马尔可夫网络刻画了一组随机变量之间的条件独立结构，即马尔可夫性质。现有工作主要关注特定分布族（如指数族）和/或特定图结构，且大多只能处理单一数据类型（连续或离散）的变量。在本工作中，我们通过广义精度矩阵（GPM）刻画了所有数据类型（包括连续、离散及混合类型）的一般分布中的条件独立结构。此外，我们允许变量间存在一般函数关系，从而在最一般的设定中提出了一种马尔可夫网络结构学习算法。为应对该问题（尤其是大规模图）的计算挑战，我们将所有情形统一纳入正则化评分匹配框架。我们验证了理论结果，并在多种设置下通过实验展示了其可扩展性。