A closed-form approach to Bayesian inference in tree-structured graphical models

TL;DR

This paper introduces a closed-form Bayesian inference method for tree-structured graphical models, enabling exact posterior computation without sampling by restricting to mixtures of spanning trees.

Contribution

It develops a novel exact inference algorithm for Bayesian structure learning in tree models using the Matrix-Tree theorem, avoiding Monte Carlo sampling.

Findings

Efficient exact computation of edge posterior probabilities.

Method performs well on synthetic and flow cytometry data.

Provides conditions for exact Bayesian inference with priors.

Abstract

We consider the inference of the structure of an undirected graphical model in an exact Bayesian framework. More specifically we aim at achieving the inference with close-form posteriors, avoiding any sampling step. This task would be intractable without any restriction on the considered graphs, so we limit our exploration to mixtures of spanning trees. We consider the inference of the structure of an undirected graphical model in a Bayesian framework. To avoid convergence issues and highly demanding Monte Carlo sampling, we focus on exact inference. More specifically we aim at achieving the inference with close-form posteriors, avoiding any sampling step. To this aim, we restrict the set of considered graphs to mixtures of spanning trees. We investigate under which conditions on the priors - on both tree structures and parameters - exact Bayesian inference can be achieved. Under these conditions, we derive a fast an exact algorithm to compute the posterior probability for an edge to belong to {the tree model} using an algebraic result called the Matrix-Tree theorem. We show that the assumption we have made does not prevent our approach to perform well on synthetic and flow cytometry data.