A0799
Title: Theoretical properties of log-concave projections in CAT(0) orthant space
Authors: Yuki Takazawa - The University of Tokyo (Japan) [presenting]
Tomonari Sei - The University of Tokyo (Japan)
Abstract: Orthant space is a space consisting of multiple nonnegative Euclidean orthants that are glued together on common faces. An important example of CAT(0) orthant spaces is the space of phylogenetic trees. One of the critical statistical challenges is the construction and estimation of probability distributions in these spaces. However, due to the complexity of the space, it is not simple to construct a parametric family of distributions. Shape-constrained density estimation is a method used to estimate distributions by imposing constraints on the shape of densities. One of the commonly used shape constraints in Euclidean spaces is log-concavity. Although the class of log-concave distributions is nonparametric, the estimation by maximum likelihood is possible. The generalization of this method to the space of phylogenetic trees has been proposed previously, and further generalization to CAT(0) orthant spaces is straightforward. This research investigates some properties of log-concave projections in CAT(0) orthant spaces. Log-concave projection refers to the log-concave density that minimizes the Kullback-Leibler divergence from a given probability measure. First, a sufficient condition is given for the existence of log-concave projections, and their uniqueness is shown. Then, by deriving a certain continuity property of a Kullback-Leibler type functional, conditions for the consistency of the maximum likelihood estimators to the log-concave projections are derived.
