  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.
