## Bernoulli

- Bernoulli
- Volume 24, Number 1 (2018), 386-407.

### The maximum likelihood threshold of a graph

Elizabeth Gross and Seth Sullivant

#### 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. We conclude by showing that these methods give exact bounds on the number of observations needed for the score matching estimator to exist with probability one.

#### Article information

**Source**

Bernoulli, Volume 24, Number 1 (2018), 386-407.

**Dates**

Received: September 2015

Revised: June 2016

First available in Project Euclid: 27 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1501142448

**Digital Object Identifier**

doi:10.3150/16-BEJ881

**Mathematical Reviews number (MathSciNet)**

MR3706762

**Zentralblatt MATH identifier**

06778333

**Keywords**

algebraic matroids Gaussian graphical models matrix completion maximum likelihood estimation maximum likelihood threshold rigidity theory score matching estimator weak maximum likelihood threshold

#### Citation

Gross, Elizabeth; Sullivant, Seth. The maximum likelihood threshold of a graph. Bernoulli 24 (2018), no. 1, 386--407. doi:10.3150/16-BEJ881. https://projecteuclid.org/euclid.bj/1501142448