Exact Expressions for the Log-likelihood's Hessian in Multivariate Continuous-Time Continuous-Trait Gaussian Evolution along a Phylogeny

TL;DR

This paper derives exact formulas for the Hessian of the likelihood in a multivariate Gaussian trait evolution model along a phylogeny, enabling precise inference and confidence region estimation.

Contribution

It provides closed-form expressions for the likelihood Hessian in a specific Gaussian evolution model, along with an R package for practical computation.

Findings

Closed-form Hessian expressions for the model

Implementation of 'glinvci' R package for inference

Supports missing data and multiple regimes

Abstract

We presents the closed form formulae for the likelihood Hessian matrix of a family of multivariate continuous-trait Gaussian Markov trait evolution model along a given phylogeny, in which the trait vector's mean is an affine function of that of its ancestor and the variance is not dependent of the trait. Accompanied with this work is an R package called 'glinvci', publicly available on The Comprehensive R Archive Network (CRAN), that can compute Hessian-based approximate confidence regions for these models while at the same time allowing users to have missing data, lost traits, and multiple evolutionary regimes.