A Note on Bayesian Networks with Latent Root Variables
This work addresses a theoretical problem in probabilistic graphical models for researchers in statistics and machine learning, but it appears incremental as it builds on existing Bayesian network frameworks.
The paper tackles the problem of characterizing the likelihood function in Bayesian networks with latent root variables, showing that the marginal distribution over manifest variables factorizes as an empirical Bayesian network and proving that the likelihood is dominated by the global maximum of this empirical model.
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.