On the Geometry of Bayesian Graphical Models with Hidden Variables
This work addresses foundational issues in Bayesian modeling for researchers, but it is incremental as it builds on existing geometric approaches without introducing new methods.
The paper investigates the geometry of likelihood in Bayesian directed graphs with hidden variables to gain insights into parameter unidentifiability, prior sensitivity, and posterior density shapes before numerical analysis, with findings applicable to more complex Bayesian networks with missing data.
In this paper we investigate the geometry of the likelihood of the unknown parameters in a simple class of Bayesian directed graphs with hidden variables. This enables us, before any numerical algorithms are employed, to obtain certain insights in the nature of the unidentifiability inherent in such models, the way posterior densities will be sensitive to prior densities and the typical geometrical form these posterior densities might take. Many of these insights carry over into more complicated Bayesian networks with systematic missing data.