Maximum Likelihood Threshold and Generic Completion Rank of Graphs

2017/03/22 by Grigoriy Blekherman, Blekherman, Grigoriy, Rainer Sinn +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Bayesian Modeling and Causal Inference #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.1703.07849

openalex publication_date 2017/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The minimum number of observations such that the maximum likelihood estimator in a Gaussian graphical model exists with probability one is called the maximum likelihood threshold of the underlying graph G. The natural algebraic relaxation is the generic completion rank introduced by Uhler. We show that the maximum likelihood threshold and the generic completion rank behave in the same way under clique sums, which gives us large families of graphs on which these invariants coincide. On the other hand we determine both invariants for complete bipartite graphs K_m,n and show that for some choices of m and n the two parameters may be quite far apart. In particular, this gives the first examples of graphs on which the maximum likelihood threshold and the generic completion rank do not agree.

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