In this work, we propose a global model selection criterion to estimate the graph of conditional dependencies of a random vector based on a finite sample. By global criterion, we mean optimizing a function over the entire set of possible graphs, eliminating the need to estimate the individual neighborhoods and subsequently combine them to estimate the graph. We prove the almost sure convergence of the graph estimator. This convergence holds provided the data is a realization of a multivariate stochastic process that satisfies a mixing condition. To the best of our knowledge, these are the first results to show the consistency of a model selection criterion for Markov random fields on graphs under non-independent data.

翻译：本文提出了一种全局模型选择准则，用于基于有限样本估计随机向量的条件依赖图。所谓全局准则，是指在整个可能图集上优化函数，无需先估计各个邻域再将其组合成图。我们证明了该图估计量的几乎必然收敛性，该收敛性成立的前提是数据为满足混合条件的多元随机过程的实现。据我们所知，这是首个在非独立数据下证明图马尔可夫随机场模型选择准则相合性的结果。