We propose a penalized pseudo-likelihood criterion to estimate the graph of conditional dependencies in a discrete Markov random field that can be partially observed. We prove the convergence of the estimator in the case of a finite or countable infinite set of variables. In the finite case the underlying graph can be recovered with probability one, while in the countable infinite case we can recover any finite sub-graph with probability one, by allowing the candidate neighborhoods to grow with the sample size n and provided the penalizing constant is sufficiently large. Our method requires minimal assumptions on the probability distribution and contrary to other approaches in the literature, the usual positivity condition is not needed. We evaluate the performance of the estimator on simulated data and we apply the methodology to a real dataset of stock index markets in different countries.

翻译：我们提出一个惩罚性的假象标准来估计离散的Markov随机字段中有条件依赖性图,可以部分观察到。我们证明在有限或可计数的无限变量组中,估计者是趋同的。在有限的情况下,基本图表可以以概率一收回,而在可计数的无限情况下,我们可以以概率一收回任何有限的子图,允许候选社区以样本大小 n 增长,只要惩罚性常数足够大。我们的方法要求对概率分布进行最低限度的假设,而与文献中的其他方法相反,我们不需要通常的假定性条件。我们用模拟数据来评估估计者的表现,我们将这种方法应用于不同国家股票指数市场的真实数据集。