This paper proposes an innovative method for constructing confidence intervals and assessing p-values in statistical inference for high-dimensional linear models. The proposed method has successfully broken the high-dimensional inference problem into a series of low-dimensional inference problems: For each regression coefficient $beta_i$, the confidence interval and $p$-value are computed by regressing on a subset of variables selected according to the conditional independence relations between the corresponding variable $X_i$ and other variables. Since the subset of variables forms a Markov neighborhood of $X_i$ in the Markov network formed by all the variables $X_1,X_2,ldots,X_p$, the proposed method is coined as Markov neighborhood regression. The proposed method is tested on high-dimensional linear, logistic and Cox regression. The numerical results indicate that the proposed method significantly outperforms the existing ones. Based on the Markov neighborhood regression, a method of learning causal structures for high-dimensional linear models is proposed and applied to identification of drug sensitive genes and cancer driver genes. The idea of using conditional independence relations for dimension reduction is general and potentially can be extended to other high-dimensional or big data problems as well.
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