# A structural Markov property for decomposable graph laws that allows control of clique intersections

## TL;DR

This paper introduces a new structural Markov property for decomposable graph laws, enabling explicit control over clique interactions and linking it to exponential family models, with implications for inference and computation.

## Contribution

It proposes a novel Markov property for decomposable graphs that allows explicit control of clique intersections, expanding modeling capabilities.

## Key findings

- Proves the equivalence of the property to an exponential family assumption.
- Discusses implications for identifiability and inference.
- Highlights computational benefits of the new property.

## Abstract

We present a new kind of structural Markov property for probabilistic laws on decomposable graphs, which allows the explicit control of interactions between cliques, so is capable of encoding some interesting structure. We prove the equivalence of this property to an exponential family assumption, and discuss identifiability, modelling, inferential and computational implications.