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 nodes. 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 as a function o(log n), with n the sample size. 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:我们提出了一个惩罚性的假象标准来估计离散的Markov随机字段中可部分观察到的有条件依赖性图。我们证明在有限或可计数的无限节点组合中,估算者是趋同的。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,我们可以通过允许候选社区作为函数o(log n)增长,并具有样本大小,来收回任何有概率的有限子图。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设性条件。我们评估模拟数据的估测员的性能,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。