# Beliefs in Markov Trees - From Local Computations to Local Valuation

## TL;DR

This paper explores the expressiveness of hypergraphs in belief propagation, showing that hyperedge valuations by conditional distributions are optimal, and introduces a method for recovering tree-structured belief networks, especially for Dempster-Shafer functions.

## Contribution

It demonstrates the limitations of hypergraph structures for belief propagation and develops a new method for recovering tree-structured belief networks from data.

## Key findings

- Hypergraph valuations by conditional distributions are optimal for belief propagation.
- Methods for recovering belief networks from data cannot find simpler hypergraph structures.
- A specialized method for Dempster-Shafer belief functions was developed.

## Abstract

This paper is devoted to expressiveness of hypergraphs for which uncertainty propagation by local computations via Shenoy/Shafer method applies. It is demonstrated that for this propagation method for a given joint belief distribution no valuation of hyperedges of a hypergraph may provide with simpler hypergraph structure than valuation of hyperedges by conditional distributions. This has vital implication that methods recovering belief networks from data have no better alternative for finding the simplest hypergraph structure for belief propagation. A method for recovery tree-structured belief networks has been developed and specialized for Dempster-Shafer belief functions