Phase-type distributions in mathematical population genetics: An emerging framework.

Area of Science:

• Population genetics
• Mathematical biology
• Probability theory

Background:

• Phase-type distributions model time to absorption in Markov chains.
• They provide a general framework for calculating properties of coalescent models.
• Key coalescent process times and spectra are phase-type distributed.

Purpose of the Study:

• To explain phase-type distribution theory.
• To demonstrate its application in deriving coalescent model properties.
• To provide tools for statistical inference in population genetics.

Main Methods:

• Utilizing matrix manipulations for analytical tractability of phase-type distributions.
• Connecting first-step analysis of coalescent models to phase-type calculations.
• Applying reward transformations for calculating covariances and correlations.

Main Results:

• Phase-type distributions simplify calculations for coalescent model properties.
• Derivation of likelihoods for small coalescent trees using phase-type theory.
• Demonstration of phase-type framework's versatility with R-code.

Conclusions:

• Phase-type distributions offer a convenient and versatile framework for understanding coalescent models.
• This approach facilitates statistical inference and provides insights into ancestral processes.
• The presented methods and R-code enable reproducible analysis.