Inference and FDR Control for Simulated Markov Random Fields in High-dimension

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.