A Note on Bayesian Networks with Latent Root Variables
Marco Zaffalon Alessandro Antonucci
We characterise the likelihood function computed from a Bayesian network with
latent variables as root nodes. We show that the marginal distribution over the
remaining, manifest, variables also factorises as a Bayesian network, which we
call empirical. A dataset of observations of the manifest variables allows us
to quantify the parameters of the empirical Bayesian net. We prove that (i) the
likelihood of such a dataset from the original Bayesian network is dominated by
the global maximum of the likelihood from the empirical one; and that (ii) such
a maximum is attained if and only if the parameters of the Bayesian network are
consistent with those of the empirical model.