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Register a new userBayesian Spanning Tree: Estimating the Backbone of the Dependence Graph
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Added by
Leo Duan
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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In multivariate data analysis, it is often important to estimate a graph characterizing dependence among (p) variables. A popular strategy uses the non-zero entries in a (ptimes p) covariance or precision matrix, typically requiring restrictive modeling assumptions for accurate graph recovery. To improve model robustness, we instead focus on estimating the {em backbone} of the dependence graph. We use a spanning tree likelihood, based on a minimalist graphical model that is purposely overly-simplified. Taking a Bayesian approach, we place a prior on the space of trees and quantify uncertainty in the graphical model. In both theory and experiments, we show that this model does not require the population graph to be a spanning tree or the covariance to satisfy assumptions beyond positive-definiteness. The model accurately recovers the backbone of the population graph at a rate competitive with existing approaches but with better robustness. We show combinatorial properties of the spanning tree, which may be of independent interest, and develop an efficient Gibbs sampler for Bayesian inference. Analyzing electroencephalography data using a Hidden Markov Model with each latent state modeled by a spanning tree, we show that results are much more interpretable compared with popular alternatives.
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We present new short proofs of known spanning tree enumeration formulae for threshold and Ferrers graphs by showing that the Laplacian matrices of such graphs admit triangular rank-one perturbations. We then characterize the set of graphs whose Laplacian matrices admit triangular rank-one perturbations as the class of special 2-threshold graphs, introduced by Hung, Kloks, and Villaamil. Our work introduces (1) a new characterization of special 2-threshold graphs that generalizes the characterization of threshold graphs in terms of isolated and dominating vertices, and (2) a spanning tree enumeration formula for special 2-threshold graphs that reduces to the aforementioned formulae for threshold and Ferrers graphs. We consider both unweighted and weighted spanning tree enumeration.
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