Abstract
To allow for principled comparison between two probabilistic graphical models defined over non-identical variable sets, they have to be lifted to a common measurable space. To this end, we propose an extension scheme for any given models and establish the formal foundation: Unmatched components are completed using conditionally uniform (Laplace) extensions such that the resulting joint distributions differ from the original ones only by multiplicative constants and coincide under projection. This preserves the probabilistic semantics while enabling the application of well-defined distributional discrepancy measures.
We establish the invariance of the induced joint under projection and use the extensions to provide a minimal structural extension of two factor graphs to the smallest common measurable space as well as to a common graphical structure by a deterministic algorithm. In addition, we discuss structural and measure-theoretic properties and identify promising criteria for comparison methodologies.
Blogger's Review: This research provides a novel theoretical foundation for comparing probabilistic graphical models, particularly in handling non-identical variable sets. By introducing Laplace extensions, the authors not only preserve the probabilistic characteristics of the models but also offer effective methods for distributional discrepancy measurement, which has significant application potential.