Tractable Bayesian Learning of Tree Belief Networks

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 ana­ lytically in polynomial time. This fol­ lows from two main results: First, we show that factored distributions over spanning trees in a graph can be inte­ grated in closed form. Second, we ex­ amine 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 com­ pactly parametrized product of Dirich­ let distributions. Besides 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 com­ puted through an ensemble average over tree structures.