The space of ultrametric phylogenetic trees

TL;DR

This paper explores the mathematical structure of ultrametric phylogenetic trees, comparing different metric spaces to understand their suitability for statistical analysis and the implications for phylogenetic inference consistency.

Contribution

It introduces and compares two natural metric spaces for ultrametric trees based on different parameterizations, highlighting the need for additional properties in choosing the appropriate space.

Findings

Few known tree space constructions satisfy formal requirements.

Basic requirements are insufficient to distinguish between the spaces.

Choice of metric space affects the summary tree and inference consistency.

Abstract

The reliability of a phylogenetic inference method from genomic sequence data is ensured by its statistical consistency. Bayesian inference methods produce a sample of phylogenetic trees from the posterior distribution given sequence data. Hence the question of statistical consistency of such methods is equivalent to the consistency of the summary of the sample. More generally, statistical consistency is ensured by the tree space used to analyse the sample. In this paper, we consider two standard parameterisations of phylogenetic time-trees used in evolutionary models: inter-coalescent interval lengths and absolute times of divergence events. For each of these parameterisations we introduce a natural metric space on ultrametric phylogenetic trees. We compare the introduced spaces with existing models of tree space and formulate several formal requirements that a metric space on phylogenetic trees must possess in order to be a satisfactory space for statistical analysis, and justify them. We show that only a few known constructions of the space of phylogenetic trees satisfy these requirements. However, our results suggest that these basic requirements are not enough to distinguish between the two metric spaces we introduce and that the choice between metric spaces requires additional properties to be considered. Particularly, that the summary tree minimising the square distance to the trees from the sample might be different for different parameterisations. This suggests that further fundamental insight is needed into the problem of statistical consistency of phylogenetic inference methods.