An empirical Bayes approach to network recovery using external knowledge

Abstract

Reconstruction of a high-dimensional network may benefit substantially from the inclusion of prior knowledge on the network topology. In the case of gene interaction networks such knowledge may come for instance from pathway repositories like KEGG, or be inferred from data of a pilot study. The Bayesian framework provides a natural means of including such prior knowledge. Based on a Bayesian Simultaneous Equation Model, we develop an appealing Empirical Bayes (EB) procedure that automatically assesses the agreement of the used prior knowledge with the data at hand. We use variational Bayes method for posterior densities approximation and compare its accuracy with that of Gibbs sampling strategy. Our method is computationally fast, and can outperform known competitors. In a simulation study, we show that accurate prior data can greatly improve the reconstruction of the network, but need not harm the reconstruction if wrong. We demonstrate the benefits of the method in an analysis of gene expression data from GEO. In particular, the edges of the recovered network have superior reproducibility (compared to that of competitors) over resampled versions of the data.

Keywords: Empirical Bayes; High-dimensional Bayesian inference; Prior information; Undirected network; Variational approximation.

Figure 1

ROC curves for BSEM (dashed) and BSEMed using perfect prior information (blue), BSEMed using 75% true edges present in the prior (brown), BSEMed using 50% true edges present in the prior (black) and BSEMed using 0% true edges present in the prior (red). Here, p = 100 and n ∈ {50, 200}.

Figure 2

Visualization of BSEMed ’${\overline{\kappa }}_{i,r}$ ’ using perfect prior (b), BSEMed ’${\overline{\kappa }}_{i,r}$ ’ using 50% true edges information (c), BSEM ’${\overline{\kappa }}_{i,r}$ ’ (d) and the true graph (a) in case n = 50 and p = 100.

Figure 3

BSEM vs BSEMed network estimates in lung cancer. Red edges are the overlap edges.

Figure 4

BSEM vs BSEMed network estimates in pancreas cancer. Red edges are the overlap edges.

Figure 5

Venn diagrams displaying the mean overlap of reproduced top-ranking edges, corresponding to the second row of Table 3 (Figure 5.a) and Table 4 (Figure 5.b).

Figure 6

Network in a normal cell vs BSEMed network in lung cancer. Red edges are the overlap edges between prior and posterior networks.

Figure 7

Network in a normal cell vs BSEMed network in pancreas cancer. Red edges are the overlap edges between prior and posterior networks.