Hadamard Conjugation for the Kimura 3ST Model: Combinatorial Proof using Pathsets

TL;DR

This paper provides a combinatorial proof for expressing probabilities in the Kimura 3ST model using Hadamard conjugation, enabling analytical computation of sequence evolution probabilities.

Contribution

It offers a direct proof of Hadamard conjugation for the Kimura 3ST model using pathset distances, expanding analytical tools in sequence evolution analysis.

Findings

Analytical expression of probabilities via Hadamard conjugation

Pathset interpretation generalizes pairwise distances

Application of Hadamard conjugation to sequence evolution problems

Abstract

In 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.