Bayesian Structure Learning in Multilayered Genomic Networks

Bayesian graphical models for modern biological applications

“…As a consequence, the graph topology and the precision matrix stay the same across observations. Motivated by the analysis of multi-platform genomics data, Ha et al (2020) proposed a Bayesian approach for chain graph selection based on node-wise likelihoods that converts the chain graph into a more tractable multiple regression model, accounting for both with and between chain component dependencies. In a chain graph, the probability distribution of the observed random variables Y can be factorized as PðYÞ ¼ Q s2T PðY s jY pa s Þ, where s represent chain components belonging to the ordered partitioning T (Lauritzen 1996a).…”

“…where B ¼ ðb vu Þ is a p  p matrix for which the zero pattern encodes the directed edges between chain components, and the precision matrix of K is a matrix for which the nonzero off-diagonal elements represent the undirected edges within a chain component after taking into account the effects from the directed edges. Ha et al (2020) derived a node-wise likelihood that, for a given node v, can be written as…”

“…where a v ¼ ÀK C v ;v =j vv , C v and P v are defined by the set of all other vertices in the same layer as v and all the preceding vertices, C tðvÞÀ1 , respectively, and e v $ Nð0; 1=j vv Þ is independent of all other random variables; see Ha et al (2020) for technical details. Within this framework, the undirected and directed edges of the chain graph can be selected using zero restrictions on the regression parameters, B and a.…”

Discussion to: Bayesian graphical models for modern biological applications by Y. Ni, V. Baladandayuthapani, M. Vannucci and F.C. Stingo

“…As clarified in Section 5.3, these covariates can play two roles: (i) they can enter directly the graphical structure of the data, giving rise to a covariate-dependent graphical model; (ii) they can enter the mean structure of the data, so that a covariate-adjusted graphical model can be used for their error term. Case (i) is covered by Section 4, where graphical regression is presented together with a few alternative techniques, while case (ii) is briefly considered in Section 5.3, featuring the method of Ha, Stingo and Ha et al (2021). A natural question is then whether the two roles could be played at the same time, in a single model, to appreciable profit.…”

Discussion to: Bayesian graphical models for modern biological applications by Y. Ni, V. Baladandayuthapani, M. Vannucci and F.C. Stingo

“…However, similar departures from normality can certainly occur in several other problems highlighted by the authors, such as in directed or multiple graphical models. A notable recent work in this direction is by Chakraborty et al (2021), who introduced the framework of sign independence of Bhadra et al (2018) into the chain graph models considered by Ha et al (2021), thereby allowing for non-normal marginals in modeling multiplatform genomic data arising from non small cell lung cancer. This is an important step, since for multi-platform genomic data that respect a chain graph hierarchy, Chakraborty et al ( 2021) fit a joint model for copy number aberrations, mRNA expressions, protein expressions and drug responses; and nonnormality can appear in multiple layers.…”