Gene network reconstruction using global-local shrinkage priors

Abstract

Reconstructing a gene network from high-throughput molecular data is an important but challenging task, as the number of parameters to estimate easily is much larger than the sample size. A conventional remedy is to regularize or penalize the model likelihood. In network models, this is often done in the neighbourhood of each node or gene. However, estimation of the many regularization parameters is often difficult and can result in large statistical uncertainties. In this paper we propose to combine local regularization with shrinkage of the regularization parameters to borrow strength between genes and improve inference. We employ a simple Bayesian model with non-sparse, conjugate priors to facilitate the use of fast variational approximations to posteriors. We discuss empirical Bayes estimation of hyper-parameters of the priors, and propose a novel approach to rank-based posterior thresholding. Using extensive model- and data-based simulations, we demonstrate that the proposed inference strategy outperforms popular (sparse) methods, yields more stable edges, and is more reproducible. The proposed method, termed ShrinkNet, is then applied to Glioblastoma to investigate the interactions between genes associated with patient survival.

Learning from a lot: Empirical Bayes for high‐dimensional model‐based prediction

“…Finding solutions and requires specific derivations for the model at hand. Several are available in the literature such as for spike‐and‐slab regression (Carbonetto & Stephens, ), the Bayesian ridge model (Leday et al, ), and the Bayesian lasso (Joo, ). For example, in the latter model is a multivariate Gaussian, whereas conveniently factorizes with respect to Z 1 ,…, Z p as a product of inverse Gaussians.…”

An empirical Bayes approach to network recovery using external knowledge

“…The algorithm is similar, but still significantly different, from the algorithm developed in Leday et al (2015) for the model (3). In the following we can see that, due to (4), the variational parameters have a form which renders the implementation of (4) much more challenging.…”

“…We developed a dedicated edge selection algorithm for BSEM model in Leday et al (2015). It is based on summarizing β i,r and β r,i by ${\overline{\kappa }}_{i,r}$, $${\overline{\kappa }}_{i,r}=\left({\kappa }_{i,r}+{\kappa }_{r,i}\right)/2\text{with}{\kappa }_{i,r}=\frac{\left|{\text{boldE}}_{q^{i*}}\left[{\beta }_{i,r}|{\text{boldy}}_i\right]\right|}{\sqrt{{\text{boldV}}_{q^{i*}}\left[{\beta }_{i,r}|{\text{boldy}}_i\right]}}$$…”

“…Previously, we proposed a Bayesian formulation of the SEM (Leday et al, 2015). In this Bayesian SEM (henceforth BSEM) the structural model (2) is endowed with the following prior: $$\begin{array}{c}left{ϵ}_i|{\sigma }_i^2,{\tau }_i^2\sim \text{N}\left(0_n,{\sigma }_i^2{\text{boldI}}_n\right),\\ {\beta }_i|{\sigma }_i^2,{\tau }_i^2\sim \text{N}\left(0_s,{\sigma }_i^2{\tau }_i^{-2}{\text{boldI}}_s\right),\\ {\tau }_i^2\sim \text{Gamma}\left(a_1,b_1\right),\\ {\sigma }_i^{-2}\sim \text{Gamma}\left(a_2,b_2\right),\end{array}$$ where I is an identity matrix, s = p − 1, and Gamma(a, b) denotes a gamma distribution with shape parameter a and rate parameter b , and ${\tau }_i^2$ and ${\sigma }_i^{-2}$ are independent.…”

“…The BSEM model can be fit computationally efficiently by a variational method, and generally outperforms the aforementioned lasso regression approach to the estimation of model (2). Furthermore, variables can be accurately selected based on the marginal posterior distributions of the regression coefficients (Leday et al, 2015). …”