Title: Maximum likelihood estimation in Gaussian models under total positivity
Abstract: The problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two (MTP2) is investigated. The maximum likelihood estimator (MLE) for such distributions exists based on just two observations, irrespective of the underlying dimension. It is further demonstrated that the MTP2 constraint serves as an implicit regularizer and leads to sparsity in the estimated inverse covariance matrix, determining what we name the ML graph. We show that the maximum weight spanning forest (MWSF) of the empirical correlation matrix is a spanning forest of the ML graph. In addition, we show that we can find an upper bound for the ML graph by adding edges to the MSWF corresponding to correlations in excess of those explained by the forest. We provide globally convergent coordinate descent algorithms for calculating the MLE under the MTP2 constraint which are structurally similar to iterative proportional scaling.
The lecture is based on recent joint work with Caroline Uhler and Piotr Zwiernik.