  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.
