  We analyze the problem of maximum likelihood estimation for Gaussian
distributions that are multivariate totally positive of order two (MTP2). By
exploiting connections to phylogenetics and single-linkage clustering, we give
a simple proof that the maximum likelihood estimator (MLE) for such
distributions exists based on at least 2 observations, irrespective of the
underlying dimension. Slawski and Hein, who first proved this result, also
provided empirical evidence showing 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 we can find an
upper bound for the ML graph by adding edges corresponding to correlations in
excess of those explained by the maximum weight spanning forest of the
correlation matrix. Moreover, we provide globally convergent coordinate descent
algorithms for calculating the MLE under the MTP2 constraint which are
structurally similar to iterative proportional scaling. We conclude the paper
with a discussion of signed MTP2 distributions.
