  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.
