Tractable Bayesian Learning of Tree Belief Networks

TL;DR

This paper introduces a family of decomposable priors for tree belief networks that enable exact Bayesian learning with complete data in polynomial time, leveraging closed-form integration of spanning tree distributions.

Contribution

It presents a novel class of priors that make Bayesian learning of tree belief networks tractable and analytically solvable, expanding the scope of efficient probabilistic modeling.

Findings

Closed-form integration of spanning tree distributions.

Priors constrained to Dirichlet distributions.

Enables exact Bayesian learning with complete data.

Abstract

In this paper we present decomposable priors, a family of priors over structure and parameters of tree belief nets for which Bayesian learning with complete observations is tractable, in the sense that the posterior is also decomposable and can be completely determined analytically in polynomial time. This follows from two main results: First, we show that factored distributions over spanning trees in a graph can be integrated in closed form. Second, we examine priors over tree parameters and show that a set of assumptions similar to (Heckerman and al. 1995) constrain the tree parameter priors to be a compactly parameterized product of Dirichlet distributions. Beside allowing for exact Bayesian learning, these results permit us to formulate a new class of tractable latent variable models in which the likelihood of a data point is computed through an ensemble average over tree structures.