## Abstract

The maximum likelihood threshold (MLT) of a graph G
is the minimum number of samples to almost surely guarantee existence
of the maximum likelihood estimate in the corresponding Gaussian
graphical model. We give a new characterization of the MLT in terms of
rigidity-theoretic properties of G and use this characterization to give new combinatorial lower bounds on the MLT of any graph.

We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{ö}s-Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial.

Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{ö}s-Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most 4 and describe the threshold probability for the MLT to switch from 3 to 4.

We also give a geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.

We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{ö}s-Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial.

Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{ö}s-Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most 4 and describe the threshold probability for the MLT to switch from 3 to 4.

We also give a geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.

Original language | English |
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Pages (from-to) | 3288-3319 |

Journal | The Annals of Applied Probability |

Volume | 34 |

Issue number | 3 |

Publication status | Published - 11 Jun 2024 |

## Keywords

- Gaussian graphical models
- Number of observations
- Maximum likelihood threshold
- Combinatorial rigidity
- Algebraic statistics