An improved shrinkage estimator to infer regulatory networks with Gaussian graphical models

Abstract

Gaussian graphical models (GGMs) are widely used to tackle the important and challenging problem of inferring genetic regulatory networks from expression data. These models have gained much attention as they encode full conditional relationships between variables, i.e. genes. As a consequence, structure learning of a GGM requires an invertible and well-conditioned covariance matrix. Unfortunately, the usual estimator---the sample covariance matrix---is ill-suited in the "small n, large p" setting characteristic of microarray data. As an alternative, [9] proposed a shrinkage estimator that is both statistically efficient and computationally fast. The effectiveness of this estimator in bioinformatics has been illustrated by [12] who successfully used it to infer genetic regulatory networks from microarray data. Unfortunately, this improved estimator requires the shrinkage intensity to be estimated from the data, which is problematic in the "small n, large p" setting. Indeed, we show that the optimal shrinkage intensity estimator used in [9, 12] is biased. We propose a parametric bootstrap approach to estimate this bias and derive a "bias-corrected" shrinkage estimator. The applicability and usefulness of our estimator are demonstrated on both simulated and real expression data.