This paper studies the consistency and statistical inference of simulated Markov random fields (MRFs) in a high dimensional background. Our estimators are based on the Markov chain Monte Carlo maximum likelihood estimation (MCMC-MLE) method, penalized by the Elastic-net. Under mild conditions that ensure a specific convergence rate of the MCMC method, the $\ell_{1}$ consistency of Elastic-net-penalized MCMC-MLE is obtained. We further propose a decorrelated score test based on the decorrelated score function and prove the asymptotic normality of the score function without the influence of many nuisance parameters under the assumption that it accelerates the convergence of the MCMC method. The one-step estimator for a single parameter of interest is constructed by linearizing the decorrelated score function to solve its root, and the normality and confidence interval for the true value, is established. We use different algorithms to control the false discovery rate (FDR) for multiple testing problems via classic p-values and novel e-values. Finally, we empirically validate the asymptotic theories and demonstrate both FDR control procedures in our article have good performance.

翻译：本文研究高维背景下模拟马尔可夫随机场（MRF）的一致性与统计推断。我们的估计量基于马尔可夫链蒙特卡洛最大似然估计（MCMC-MLE）方法，并采用弹性网络（Elastic-net）进行惩罚。在确保MCMC方法具有特定收敛速率的温和条件下，我们得到了弹性网络惩罚的MCMC-MLE的ℓ1一致性。我们进一步基于去相关得分函数提出了去相关得分检验，并证明了在假设其加速MCMC方法收敛的前提下，得分函数在不受到众多干扰参数影响下的渐近正态性。通过线性化去相关得分函数以求解其零点，我们构造了单个感兴趣参数的一步估计量，并建立了真实值的正态性与置信区间。我们利用经典p值和新型e值，采用不同算法控制多重检验问题中的错误发现率（FDR）。最后，我们通过实证验证了渐近理论，并展示了本文中两种FDR控制程序均具有良好的性能。