2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101458In most stochastic models of molecular sequence evolution the probability of each possible pattern of homologous characters at a site is estimated numerically. However in the case of Kimura's three-substitution-types (K3ST) model, these probabilities can be expressed analytically by Hadamard conjugation as a function of the phylogeny T and the substitution probabilities on each edge of T, together with an analytic inverse function. In this paper we produce a direct proof of these results, using pathset distances which generalise pairwise distances between sequences. This interpretation allows us to apply Hadamard conjugation to a number of topical problems in the mathematical analysis of sequence evolution.Populations and Evolutiontext