The Cauchy Process on Phylogenies: A Tractable Model for Pulsed Evolution.

Area of Science:

• Evolutionary biology
• Phylogenetics
• Computational biology

Background:

• Phylogenetic comparative methods model trait evolution using random processes like Brownian Motion.
• Non-gradual evolution necessitates complex models, often Lévy processes, but their inference is computationally intensive.
• Existing methods for complex models rely on approximations, high-dimensional sampling, or numerical integration.

Purpose of the Study:

• To introduce an exact algorithm for the Cauchy Process (CP), a pure-jump Lévy process, for phylogenetic comparative analysis.
• To enable efficient statistical inference of trait evolution under the CP, including joint probability densities and ancestral trait reconstructions.
• To provide a computationally feasible and accurate method for modeling pulsed or non-gradual evolutionary patterns.

Main Methods:

• Developed an exact algorithm to compute the joint probability density of tip trait values under a CP.
• Derived methods for computing posterior densities of ancestral trait values and branch increments in quadratic time.
• Implemented the algorithm in C with an R interface (package 'cauphy').

Main Results:

• The CP generates distinct evolutionary patterns compared to Gaussian processes.
• Restricted maximum likelihood parameter estimates and root trait reconstructions are unbiased and accurate for trees up to 200 tips.
• The CP model can reconstruct multimodal posterior ancestral trait distributions, reflecting evolutionary uncertainty.

Conclusions:

• The Cauchy Process offers a computationally efficient and statistically robust method for modeling complex, non-gradual trait evolution.
• The developed exact algorithm provides accurate parameter estimation and ancestral state reconstruction.
• The 'cauphy' R package facilitates the application of the CP to diverse datasets in evolutionary ecology and virology.