Logarithmic Voronoi Cells

A statistical model is a subset of a probability simplex. The maximum likelihood estimation (MLE) is a map that takes an empirical data point and assigns it a point in the model that maximizes the log-likelihood function defined by the data point. Finding the maximum likelihood estimate for a given data point is a problem that can be solved using optimization tools. On the other hand, given a point in the model, we may study the fiber of the MLE map corresponding to that point, known as its logarithmic Voronoi cell. Each logarithmic Voronoi cell lives inside its log-normal polytope, and these log-normal polytopes corresponding to the points of our model fill the probability simplex. We introduce these notions and describe the situation geometrically using examples.