# Phylogenetic Factor Analysis

## TL;DR

This paper introduces phylogenetic factor analysis (PFA), a Bayesian method that simplifies high-dimensional trait evolution modeling by identifying independent factors, improving computational efficiency and model fit in evolutionary studies.

## Contribution

The paper presents PFA, a novel Bayesian approach that infers evolutionary factors, handles mixed data types, and incorporates phylogenetic uncertainty with efficient Gibbs sampling.

## Key findings

- PFA outperforms multivariate diffusion in computational speed.
- PFA provides better model fit in evolutionary case studies.
- PFA effectively integrates missing data and discrete traits.

## Abstract

Phylogenetic comparative methods explore the relationships between quantitative traits adjusting for shared evolutionary history. This adjustment often occurs through a Brownian diffusion process along the branches of the phylogeny that generates model residuals or the traits themselves. For high-dimensional traits, inferring all pair-wise correlations within the multivariate diffusion is limiting. To circumvent this problem, we propose phylogenetic factor analysis (PFA) that assumes a small unknown number of independent evolutionary factors arise along the phylogeny and these factors generate clusters of dependent traits. Set in a Bayesian framework, PFA provides measures of uncertainty on the factor number and groupings, combines both continuous and discrete traits, integrates over missing measurements and incorporates phylogenetic uncertainty with the help of molecular sequences. We develop Gibbs samplers based on dynamic programming to estimate the PFA posterior distribution, over three-fold faster than for multivariate diffusion and a further order-of-magnitude more efficiently in the presence of latent traits. We further propose a novel marginal likelihood estimator for previously impractical models with discrete data and find that PFA also provides a better fit than multivariate diffusion in evolutionary questions in columbine flower development, placental reproduction transitions and triggerfish fin morphometry.