The Maximum Likelihood Threshold of a Graph

TL;DR

This paper explores the maximum likelihood threshold in Gaussian graphical models, revealing its connection to combinatorial rigidity and providing bounds based on matroid theory.

Contribution

It establishes a novel link between maximum likelihood threshold and combinatorial rigidity, enabling new bounds and insights for Gaussian graphical models.

Findings

Maximum likelihood threshold is connected to the rigidity matroid.

If edges form an independent set in the rigidity matroid, the threshold is bounded by n.

The paper provides new bounds and results for the threshold based on rigidity theory.

Abstract

The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an independent set in the $n-1$-dimensional generic rigidity matroid, then the maximum likelihood threshold of $G$ is less than or equal to $n$. This connection allows us to prove many results about the maximum likelihood threshold.