On the Wald Space for Phylogenetic Trees

Most existing metrics between phylogenetic trees directly measure differences in topology and edge weights, and are unrelated to the models of evolution used to infer trees. We describe metrics which instead are based on distances between the probability models of discrete or continuous characters induced by trees. We describe how construction of information-based geodesics leads to the recent (Garba et al. in J Math Biol 82(3):1–39, 2021, [1]) wald space of phylogenetic trees. As a point set, it sits between the BHV space (Billera et al. in Adv Appl Math 27(4):733–767, 2001, [2]) and the edge-product space (Moulton and Steel in Adv Appl Math 33(4):710–727, 2004, [3]). It has a natural embedding into the space of symmetric positive definite matrices, equipped with the information geometry. Thus, singularities such as overlapping leaves are infinitely far away, proper forests, however, comprising the “BHV-boundary at infinity”, are part of the wald space, adding boundary correspondences to groves (corresponding to orthants in the BHV space). In fact, the wald space contracts to the completely disconnected forest. Further, it is a geodesic space, exhibiting the structure of a Whitney stratified space of type (A) where strata carry compatible Riemannian metrics. We explore some more geometric properties, but the full picture remains open. We conclude by identifying open problems we deem interesting (Lueg et al. in J Lond Math Soc 109(5):e12893, 2024, [4]).