Tropical Foundations for Probability & Statistics on Phylogenetic Tree Space

Abstract

We introduce a novel framework for the statistical analysis of phylogenetic trees: Palm tree space is con- structed on principles of tropical algebraic geometry, and represents phylogenetic trees as a point in a space endowed with the tropical metric. We show that palm tree space possesses a variety of properties that allow for the definition of probability measures, and thus expectations, variances, and other fundamental statistical quantities. This provides a new, tropical basis for a statistical treatment of evolutionary biological processes represented by phylogenetic trees. In particular, we show that a geometric approach to phylogenetic tree space — first introduced by Billera, Holmes, and Vogtmann, which we reinterpret in this paper via tropical geometry — results in analytic, geometric, and topological characteristics that are desirable for probability, statistics, and increased computational efficiency.