Tail paradox, partial identifiability, and influential priors in Bayesian branch length inference

Area of Science:

• Computational Biology
• Phylogenetics
• Statistical Modeling

Background:

• Bayesian phylogenetic analyses, particularly using MrBayes, can yield excessively large branch lengths.
• This phenomenon, where posterior credibility intervals exclude maximum likelihood estimates, has been attributed to issues like poor convergence, mixing problems, and misspecified priors.

Purpose of the Study:

• To investigate the behavior of Bayesian Markov chain Monte Carlo (MCMC) algorithms in the tail of posterior distributions.
• To identify the primary causes of extreme branch length estimates in Bayesian phylogenetics.
• To propose improved prior specifications for branch lengths.

Main Methods:

• Analysis of Bayesian MCMC algorithm behavior with infinite branch lengths.
• Evaluation of the impact of default priors on branch length estimation.
• Development and testing of novel multivariate priors (compound Dirichlet priors) on a star phylogeny.

Main Results:

• The likelihood function in Bayesian phylogenetics approaches a constant as branch lengths increase, leading to poor mixing and undue prior influence.
• Default priors in programs like MrBayes, which assume independent and identical distributions for branch lengths, are identified as a major cause of extreme estimates.
• The proposed compound Dirichlet priors demonstrate utility in estimating branch lengths, especially on star phylogenies.

Conclusions:

• The choice of prior distribution for branch lengths is critical in Bayesian phylogenetics.
• Default priors in current software can lead to unreliable branch length estimates.
• Compound Dirichlet priors offer a more robust and flexible alternative for high-dimensional prior specification in Bayesian analyses.