Selection and estimation for mixed graphical models.
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.
Department of Biostatistics, University of Washington, Box 357232, Seattle, Washington 98195, U.S.A.
Shizhe
S
Chen
Department of Biostatistics, University of Washington, Box 357232, Seattle, Washington 98195, U.S.A.
Daniela M
DM
Witten
Department of Biostatistics, University of Washington, Box 357232, Seattle, Washington 98195, U.S.A.
Ali
A
Shojaie
Compatibility; Conditional likelihood; Exponential family; High dimensionality; Model selection consistency; Neighbourhood selection; Pairwise Markov random field