Selection and estimation for mixed graphical models

Abstract

Summary We consider the problem of estimating the parameters in a pairwise graphical model in which the distribution of each node, conditioned on the others, may have a different exponential family form. We identify restrictions on the parameter space required for the existence of a well-defined joint density, and establish the consistency of the neighbourhood selection approach for graph reconstruction in high dimensions when the true underlying graph is sparse. Motivated by our theoretical results, we investigate the selection of edges between nodes whose conditional distributions take different parametric forms, and show that efficiency can be gained if edge estimates obtained from the regressions of particular nodes are used to reconstruct the graph. These results are illustrated with examples of Gaussian, Bernoulli, Poisson and exponential distributions. Our theoretical findings are corroborated by evidence from simulation studies.

High Dimensional Semiparametric Latent Graphical Model for Mixed Data

“…() and Chen et al . () proposed exponential family graphical models, which allow the conditional distribution of nodes to belong to the exponential family. Later, a semiparametric exponential family graphical model was studied by Yang et al .…”

“…() and Chen et al . () essentially models the nodewise conditional distribution by generalized linear models. In contrast, the latent Gaussian copula model is a generative model which combines continuous and discrete data through a deeper layer of unobserved variables.…”

Replicates in high dimensions, with applications to latent variable graphical models

“…By an application of Proposition 1 in Chen et al (2015), under these conditions, there exists a joint probability distribution for the model introduced in Definition 1 that takes the form $$pfalse(xfalse)\propto \text{exp}false\{\frac{1}{2}\sum _{j=1}^{p+h}\sum _{k\ne j}{\beta }_{jk}{x}_j{x}_k+\sum _{j=1}^{p+h}{f}_{j}false(x_{j}false)false\}.$$…”

“…The Gaussian graphical model has been studied extensively (Meinshausen & Bühlmann, 2006; Yuan & Lin, 2007; Friedman et al, 2008; Rothman et al, 2008; Peng et al, 2009; Ravikumar et al, 2011; Cai et al, 2011; Sun & Zhang, 2013). Other authors have considered extensions to the case in which each node-conditional distribution belongs to a univariate exponential family (Ravikumar et al, 2010; Yang et al, 2015; Lee & Hastie, 2015; Chen et al, 2015). Others have considered estimating conditional dependence relationships using semiparametric or nonparametric approaches (Liu et al, 2009, 2012; Fellinghauer et al, 2013; Voorman et al, 2014).…”

An empirical Bayes approach to network recovery using external knowledge

“…Our choice for SEM is mainly motivated by its flexibility and its performance. It can account for experimental or biological covariates in the regression, and extensions to non-Gaussian data are available (Chen et al, 2015; Allen and Liu, 2013; Yang et al, 2012; Ravikumar et al, 2010). Its Bayesian counterpart is appealing for including prior knowledge, which likely is more complicated in many other frameworks.…”