# PhaseGen: exact solutions for time-inhomogeneous multivariate coalescent distributions under diverse demographies

## TL;DR

PhaseGen is a new tool that allows precise modeling of complex population histories using advanced mathematical methods.

## Contribution

The study introduces a novel application of time-inhomogeneous phase-type theory to model diverse demographic scenarios with exact computations.

## Key findings

- PhaseGen enables exact computation of coalescent moments under piecewise-constant demographies.
- The method supports modeling complex events like population expansions and bottlenecks.
- Gradient-based optimization is now possible for fitting demographic models to data.

## Abstract

Phase-type theory is emerging as a powerful framework for modeling coalescent processes, allowing for the exact computation of quantities of interest. This includes moments of tree height, total branch length, the site-frequency spectrum, and the full distribution of the time to the most recent common ancestor. However, prior applications have largely been limited to time-homogeneous settings, with constant population sizes and migration rates, restricting the range of demographic scenarios that can be modeled. In this study, we apply time-inhomogeneous phase-type theory to enable the exact computation of (cross-)moments of arbitrary order and reward structure under piecewise-constant demographies. This extension enables the modeling of significantly more complex demographic scenarios, including population expansions, contractions, bottlenecks, and splits. It furthermore supports fitting demographic models to data through gradient-based optimization. To support these advancements, we introduce PhaseGen—a software package designed for the numerically stable computation of exact moments under diverse demographic scenarios, with support for gradient-based parameter estimation.