Online Bayesian phylogenetic inference: theoretical foundations via Sequential Monte Carlo

TL;DR

This paper develops a theoretical foundation for online Bayesian phylogenetic inference using Sequential Monte Carlo, enabling efficient updates of evolutionary trees as new genetic data becomes available.

Contribution

It provides the first theoretical analysis of online Bayesian phylogenetics, including consistency results and bounds on likelihood surface changes with new data.

Findings

SMC methods are consistent in sampling from the correct distribution.

Effective sample size (ESS) grows linearly with the number of particles.

Bounds on likelihood surface changes facilitate performance analysis.

Abstract

Phylogenetics, the inference of evolutionary trees from molecular sequence data such as DNA, is an enterprise that yields valuable evolutionary understanding of many biological systems. Bayesian phylogenetic algorithms, which approximate a posterior distribution on trees, have become a popular if computationally expensive means of doing phylogenetics. Modern data collection technologies are quickly adding new sequences to already substantial databases. With all current techniques for Bayesian phylogenetics, computation must start anew each time a sequence becomes available, making it costly to maintain an up-to-date estimate of a phylogenetic posterior. These considerations highlight the need for an \emph{online} Bayesian phylogenetic method which can update an existing posterior with new sequences. Here we provide theoretical results on the consistency and stability of methods for online Bayesian phylogenetic inference based on Sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). We first show a consistency result, demonstrating that the method samples from the correct distribution in the limit of a large number of particles. Next we derive the first reported set of bounds on how phylogenetic likelihood surfaces change when new sequences are added. These bounds enable us to characterize the theoretical performance of sampling algorithms by bounding the effective sample size (ESS) with a given number of particles from below. We show that the ESS is guaranteed to grow linearly as the number of particles in an SMC sampler grows. Surprisingly, this result holds even though the dimensions of the phylogenetic model grow with each new added sequence.