Maximum Likelihood Estimation of Log-Concave Densities on Tree Space

TL;DR

This paper develops a maximum likelihood estimation method for log-concave densities on the space of phylogenetic trees, providing theoretical guarantees and demonstrating improved estimation accuracy over kernel methods.

Contribution

It establishes conditions for the existence and uniqueness of the estimator on tree space and proposes an efficient estimation algorithm for low dimensions.

Findings

Estimator has smaller integrated squared error for large samples.

The method works well for log-concave true densities.

Numerical experiments show effective clustering performance.

Abstract

Phylogenetic trees are key data objects in biology, and the method of phylogenetic reconstruction has been highly developed. The space of phylogenetic trees is a nonpositively curved metric space. Recently, statistical methods to analyze the set of trees on this space are being developed utilizing this property. Meanwhile, in Euclidean space, the log-concave maximum likelihood method has emerged as a new nonparametric method for probability density estimation. In this paper, we derive a sufficient condition for the existence and uniqueness of the log-concave maximum likelihood estimator on tree space. We also propose an estimation algorithm for one and two dimensions. Since various factors affect the inferred trees, it is difficult to specify the distribution of sample trees. The class of log-concave densities is nonparametric, and yet the estimation can be conducted by the maximum likelihood method without selecting hyperparameters. We compare the estimation performance with a previously developed kernel density estimator numerically. In our examples where the true density is log-concave, we demonstrate that our estimator has a smaller integrated squared error when the sample size is large. We also conduct numerical experiments of clustering using the Expectation-Maximization (EM) algorithm and compare the results with k-means++ clustering using Fr\'echet mean.