[Paper Review] Marginal likelihoods in phylogenetics: a review of methods and applications

Jamie R. Oaks, Kerry A. Cobb+2 more|arXiv (Cornell University)|May 10, 2018

Evolution and Paleontology StudiesEarth and Planetary Sciences85 references4 citations

TL;DR

This paper reviews methods for estimating marginal likelihoods in Bayesian phylogenetics, emphasizing their role in model comparison and evolutionary inference. It evaluates techniques like thermodynamic integration and ABC-GLM, finding ABC-GLM biased despite good posterior estimates, and highlights challenges and future directions in Bayesian model choice for complex phylogenetic models.

ABSTRACT

By providing a framework of accounting for the shared ancestry inherent to all life, phylogenetics is becoming the statistical foundation of biology. The importance of model choice continues to grow as phylogenetic models continue to increase in complexity to better capture micro and macroevolutionary processes. In a Bayesian framework, the marginal likelihood is how data update our prior beliefs about models, which gives us an intuitive measure of comparing model fit that is grounded in probability theory. Given the rapid increase in the number and complexity of phylogenetic models, methods for approximating marginal likelihoods are increasingly important. Here we try to provide an intuitive description of marginal likelihoods and why they are important in Bayesian model testing. We also categorize and review methods for estimating marginal likelihoods of phylogenetic models, highlighting several recent methods that provide well-behaved estimates. Furthermore, we review some empirical studies that demonstrate how marginal likelihoods can be used to learn about models of evolution from biological data. We discuss promising alternatives that can complement marginal likelihoods for Bayesian model choice, including posterior-predictive methods. Using simulations, we find one alternative method based on approximate-Bayesian computation (ABC) to be biased. We conclude by discussing the challenges of Bayesian model choice and future directions that promise to improve the approximation of marginal likelihoods and Bayesian phylogenetics as a whole.

Motivation & Objective

• To provide an intuitive understanding of marginal likelihoods and their importance in Bayesian model comparison within phylogenetics.
• To review and categorize existing methods for approximating marginal likelihoods in complex phylogenetic models.
• To evaluate the performance of alternative methods, including approximate Bayesian computation (ABC), in estimating marginal likelihoods.
• To demonstrate empirical applications of marginal likelihoods in learning about evolutionary processes from biological data.
• To identify challenges in Bayesian model choice and suggest future research directions for improving marginal likelihood estimation.

Proposed method

• Uses thermodynamic integration and path sampling to estimate marginal likelihoods via MCMC sampling across temperature bridges.
• Applies quadrature integration with rectangular and trapezoidal rules to compute 'true' marginal likelihoods as reference values.
• Employs approximate Bayesian computation (ABC) with a sufficient summary statistic (proportion of variable sites) to estimate posterior densities.
• Uses ABC-GLM (approximate Bayesian computation with generalized linear models) to estimate marginal densities from ABC samples with a bandwidth of 0.002.
• Compares ABC-GLM estimates to full-likelihood MCMC results to assess approximation error, using branch length as a key parameter.
• Validates results across 100 simulated datasets under both correct and vague priors to ensure robustness and reproducibility.

Experimental results

Research questions

• RQ1 How well do different methods estimate marginal likelihoods in phylogenetic models with complex parameter spaces?
• RQ2 What is the performance of ABC-GLM relative to full-likelihood MCMC in estimating marginal likelihoods and Bayes factors?
• RQ3 How do marginal likelihoods facilitate model comparison in evolutionary biology, especially in hierarchical and high-dimensional models?
• RQ4 What are the sources of bias in ABC-based marginal likelihood estimation, and how can they be detected and mitigated?
• RQ5 What are the key challenges and future directions in improving the accuracy and efficiency of marginal likelihood estimation in Bayesian phylogenetics?

Key findings

• Quadrature integration with 1,000 and 10,000 steps produced identical marginal likelihood estimates to at least five decimal places across all 100 datasets, confirming numerical stability.
• ABC-GLM estimates of marginal likelihoods were found to be biased when compared to quadrature-based reference values, despite accurate posterior estimates.
• The ABC-GLM method produced nearly identical branch length estimates to full-likelihood MCMC, indicating minimal error from the tolerance approximation.
• The study confirms that marginal likelihoods provide a natural penalty for model complexity, favoring models that balance fit and parsimony.
• Posterior-predictive methods and alternative approaches to marginal likelihoods are promising but require further development to ensure reliability.
• The results underscore the importance of using well-calibrated methods for marginal likelihood estimation, as biased approximations can mislead model choice.