Completions to Discrete Probability Distributions in Log-linear Models

Completion problems, of recovering points 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, of determining those observed probabilities which can be finitely completed to a probability distribution in a given log--linear model. These observed probabilities either have a unique completion or two completions to the log--linear model depending on the set of observed coordinates.