Tropical Foundations for Probability & Statistics on Phylogenetic Tree Space
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We introduce a novel framework for the statistical analysis of phylogenetic trees: Palm tree space is constructed 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.
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