Completions to Discrete Probability Distributions in Log-linear Models

Completion problems, of recovering a point from a set of observed coordinates, are abundant in applications to image reconstruction, phylogenetics, and data science. We consider a completion problem coming from algebraic statistics: to describe the completions of a point to a probability distribution lying in a given log-linear model. When there are finitely many completions, we show that these points either have a unique completion or two completions to the log-linear model depending on the set of observed coordinates. We additionally describe the region of points which have a completion to the log-linear model.