We propose a penalized conditional likelihood criterion to estimate the basic neighborhood of each node 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. The estimated neighborhoods can be combined to estimate the underlying graph. In the finite case, the graph can be recovered with probability one. In contrast, we can recover any finite subgraph with probability one in the countable infinite case by allowing the candidate neighborhoods to grow as a function , with 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 estimator's performance on simulated data and apply the methodology to a real dataset of stock index markets in different countries.

中文翻译：

---

不一定为正分布的图上部分观察到的离散马尔可夫随机场的结构恢复

我们提出了一种惩罚条件似然准则来估计可以部分观察的离散马尔可夫随机场中每个节点的基本邻域。我们证明了估计器在有限或可数无限节点集的情况下的收敛性。可以组合估计的邻域来估计基础图。在有限情况下，图可以以概率 1 恢复。相反，我们可以通过允许候选邻域作为函数增长来恢复可数无限情况下概率为 1 的任何有限子图， 和样本大小。我们的方法需要对概率分布进行最少的假设，并且与文献中的其他方法相反，不需要通常的正性条件。我们评估估计器在模拟数据上的表现，并将该方法应用于不同国家股票指数市场的真实数据集。