Quantifying MCMC Exploration of Phylogenetic Tree Space

Abstract

In order to gain an understanding of the effectiveness of phylogenetic Markov chain Monte Carlo (MCMC), it is important to understand how quickly the empirical distribution of the MCMC converges to the posterior distribution. In this article, we investigate this problem on phylogenetic tree topologies with a metric that is especially well suited to the task: the subtree prune-and-regraft (SPR) metric. This metric directly corresponds to the minimum number of MCMC rearrangements required to move between trees in common phylogenetic MCMC implementations. We develop a novel graph-based approach to analyze tree posteriors and find that the SPR metric is much more informative than simpler metrics that are unrelated to MCMC moves. In doing so, we show conclusively that topological peaks do occur in Bayesian phylogenetic posteriors from real data sets as sampled with standard MCMC approaches, investigate the efficiency of Metropolis-coupled MCMC (MCMCMC) in traversing the valleys between peaks, and show that conditional clade distribution (CCD) can have systematic problems when there are multiple peaks.

Ricci-Ollivier Curvature of the Rooted Phylogenetic Subtree-Prune-Regraft Graph

“…As we prove in Lemma 6.7, the notions of coarse and asymptotic curvature differ only by a small factor bounded by It is necessary to have an efficient method of constructing the full rSPR graph for a fixed number of leaves in order to study it. The previous best algorithm for this problem requires O(m 2 n) time, where m is the number of trees in the graph and n the number of leaves [44]. Here we reduce that time to O(mn 3 ).…”

“…In previous work [44], we constructed (unrooted) SPR graphs from subsets of m high probability trees sampled from phylogenetic posteriors to compare mixing and identify local maxima. Although the SPR distance (rooted and unrooted) is NP-hard to compute [3,12], it is fixed-parameter tractable with respect to the distance in the rooted case [3].…”

“…Although the SPR distance (rooted and unrooted) is NP-hard to compute [3,12], it is fixed-parameter tractable with respect to the distance in the rooted case [3]. In particular, one can determine in O(n)-time whether two rooted phylogenetic trees are adjacent in the rSPR graph (O(n 2 )-time for unrooted trees) using the algorithms of Whidden et al [42][43][44]46]. We applied this method comparing each of the m trees pairwise to identify adjacencies, requiring a total of O(m 2 n)-time (O(m 2 n 2 )-time in the unrooted case).…”

Efficiently Inferring Pairwise Subtree Prune-and-Regraft Adjacencies between Phylogenetic Trees

“…Maximization methods aim to find the "best" tree according to an optimization criteria such as likelihood [13,16] or parsimony [17], while Bayesian statistical methods [14,3] aim to efficiently sample trees. In both cases the topology of the trees is the most difficult parameter to optimize or sample [11,9,20]. Applying tree-modifying moves in the process of maximization or sampling can be thought of as traversing the graph consisting of trees as vertices and moves as edges.…”

“…This is no trivial task, as it is NP-hard to even determine the minimum distance between a pair of trees in terms of NNI [5], SPR [2,8], or TBR [1] operations. Thus we propose: Figure 2: Two SPR graphs of high-probability tree posterior subsets from [20]. Node size indicates posterior probability.…”