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Probabilistic Graphical Model Representation in Phylogenetics

Sebastian Höhna, Tracy A. Heath, Bastien Boussau, Michael J. Landis, Fredrik Ronquist, John P. Huelsenbeck

arXiv:1312.2386v1q-bio.PE

TL;DR

Phylogenetic models vary widely, and complex graphs can become overwhelming to visualize. The paper extends graphical-model representations with tree plates and modular components; these capture tree-topology learning and allow large model families to be represented compactly. The authors also identify a boundary: BUGS lacks several capabilities needed for phylogenetic graphical models.

  • Problem

    Visualizing complex phylogenetic models can become overwhelming, even when tree plates simplify their representation.

    Source: Phylogenetic Model Graphs · p. 15

  • Method

    The paper introduces tree plates for tree-structured variables and modular representations that divide complex models at pivot nodes.

    Source: Discussion · pp. 34–35

  • Results

    Tree plates represent large trees while capturing topology inference, and combinations of modules compactly represent many models.

    Source: Discussion · pp. 34–35

  • Takeaways & Limitations

    Graphical-model algorithms may provide useful computational connections for phylogenetics as its models grow more complex.

    Source: Discussion · p. 35

    BUGS lacks tree topologies, instantaneous rate matrices, and continuous-time Markov chains, and does not handle phylogenetic graph-structure learning.

    Source: Software Implementation · p. 36

Abstract

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Recent years have seen a rapid expansion of the model space explored in statistical phylogenetics, emphasizing the need for new approaches to statistical model representation and software development. Clear communication and representation of the chosen model is crucial for: (1) reproducibility of an analysis, (2) model development and (3) software design. Moreover, a unified, clear and understandable framework for model representation lowers the barrier for beginners and non-specialists to grasp complex phylogenetic models, including their assumptions and parameter/variable dependencies. Graphical modeling is a unifying framework that has gained in popularity in the statistical literature in recent years. The core idea is to break complex models into conditionally independent distributions. The strength lies in the comprehensibility, flexibility, and adaptability of this formalism, and the large body of computational work based on it. Graphical models are well-suited to teach statistical models, to facilitate communication among phylogeneticists and in the development of generic software for simulation and statistical inference. Here, we provide an introduction to graphical models for phylogeneticists and extend the standard graphical model representation to the realm of phylogenetics. We introduce a new graphical model component, tree plates, to capture the changing structure of the subgraph corresponding to a phylogenetic tree. We describe a range of phylogenetic models using the graphical model framework and introduce modules to simplify the representation of standard components in large and complex models. Phylogenetic model graphs can be readily used in simulation, maximum likelihood inference, and Bayesian inference using, for example, Metropolis-Hastings or Gibbs sampling of the posterior distribution.

An Introduction to Probabilistic Graphical Models

The introduction uses graphical models to make phylogenetic assumptions and dependencies visible, then extends standard graph notation to represent trees and repeated characters. Examples range from a simple binary trait to sequence evolution, while showing where explicit graphs become unwieldy.

An Introduction to Probabilistic Graphical Models

A non-phylogenetic presence/absence model

A simple phylogenetic model

  • The phylogenetic binary-trait model makes each node state depend on its parent, branch length, and equilibrium frequency θ, with a separate root probability p.The illustrated tree is treated as known, and the supplementary analysis reports a 50/50 chance that the mammalian ancestor had a baculum.Source: A simple phylogenetic model · pp. 7–8, Figure 3 · p. 9

An Introduction to Probabilistic Graphical Models

  • Explicit graphs become impractical for many characters or complex models, motivating factorization into more manageable components.The simple example’s graph is already described as busy.Source: A simple phylogenetic model · p. 8

Using plates to represent repetition in the graph

A general-time-reversible model for sequence evolution

  • The GTR example represents Q as a deterministic function of exchangeability rates and base frequencies; estimated branch lengths receive exponential priors.The tree is replicated across sequence sites, and the examples retain a recognizable underlying tree structure.Source: A general-time-reversible model for… · p. 12, Figure 5 · p. 13

Phylogenetic Model Graphs

Tree plates capture the recursive, topology-dependent structure of phylogenetic models without requiring a separate graph for each tree. Pivot-based modules then make complex models easier to summarize at a high level or inspect in detail.

Phylogenetic Model Graphs

  • Tree plates represent phylogenetic recursion with root, internal, and tip nodes, and allow the graph structure to change with the tree topology.The tree variable supplies the structure; additional node classes are needed when models distinguish particular nodes or subtrees.Source: Phylogenetic Model Graphs · pp. 15–16
  • Because a tree plate uses three node classes for most models, its recursive representation avoids clutter as the number of tips grows.It can represent any tree topology; only the tree variable’s value reveals the actual graphical structure.Source: Phylogenetic Model Graphs · pp. 14–15
  • Modules can be collapsed for a compact overview or expanded to show component subgraphs, supporting both whole-model communication and focused detail.For related models, common modules can be described once and variants characterized by their differing subgraphs.Source: Phylogenetic Model Graphs · p. 17
  • Pivot nodes let a single random variable appear in multiple subgraphs, defining how upstream and downstream modules connect.The same variable names and plate indices make connections explicit when modules are expanded.Source: Phylogenetic Model Graphs · p. 17
  • The proposed modules cover tree priors, rate matrices, site-rate variation, branch rates or lengths, and the core PhyloCTMC process.Rate matrices may be shared or replicated across branches or sites; site rates can use gamma, mixture, or other distributions.Source: Phylogenetic Model Graphs · pp. 19, 21, 22
  • A species-tree–gene-tree example represents gene-level replication on a plate, with gene trees depending on a shared species tree.The PhyloCTMC component is clamped to observed sequences at the leaves.Source: Phylogenetic Model Graphs · p. 22, Figure 9 · p. 23

Computations on Model Graphs

Model graphs use conditional-independence structure to factor joint probabilities and support inference over phylogenetic variables. The section connects general graphical-model algorithms to tree-based and broader phylogenetic computations, including efficient sampling and simulation.

Joint probability

  • Conditional-independence structure factors the joint probability into node-level conditional distributions given each node’s parents.This factorization supports convenient and efficient computation.Source: Factorization · p. 25

Conditional and Marginal Distribution

  • Variable elimination computes query-node probabilities by eliminating remaining variables, but its cost depends on elimination order and graph structure.Naive summation over |R| variables with k states each requires k^|R| terms; optimal ordering is NP-hard in general, though known for common graph types such as trees.Source: Conditional and Marginal Distribution · pp. 26–27

Sum-Product Algorithm and Belief Propagation

Factor graphs and extensions

  • Factor graphs make probability distributions explicit as factor nodes, and belief propagation can extend beyond tree graphs to some other phylogenetic computations.The paper describes its use for mixture components, including rate categories and tree topologies.Source: Sum-Product Algorithm and Belief… · pp. 29, 31, Figure 11 · p. 30

MCMC sampling

  • Graphical-model factorization lets Metropolis-Hastings updates focus on changed stochastic nodes and their children rather than recomputing the full model probability.The paper notes that the resulting probability-ratio computation is faster and independent of the number of taxa for its baculum example.Source: MCMC Sampling · pp. 32–33

Sampling and simulation

  • Graph structure also supports Gibbs sampling when a variable’s prior is conjugate to its conditional posterior, and simulation by traversing from source nodes to sink nodes.Unclamping all nodes permits MCMC draws from the full model; graph traversal can generate independent samples.Source: MCMC Sampling · p. 33

Discussion

Graphical models are proposed as a standardized way to describe complex phylogenetic models, making assumptions and dependencies explicit. Tree plates and modularization address phylogenetic graph structure and scale, while the framework also connects phylogenetics to established statistical and computational methods.

Discussion

  • The paper argues that graphical models have not previously been used in phylogenetics, despite earlier suggestions, and proposes them to address challenges from model complexity.The authors also note practical software challenges in representing large PhyloGMs efficiently.Source: Discussion · pp. 34–36
  • Tree plates represent large evolutionary trees whose topology is a random variable, capturing the structure-learning aspect of phylogenetic inference.They allow efficient representation of trees with many tips.Source: Discussion · p. 34
  • Modularization splits complex phylogenetic graphs at pivot nodes, enabling compact high-level views and detailed views of selected subgraphs.Combining modules in different patterns can represent a large set of models efficiently.Source: Discussion · p. 35
  • Graphical models make assumptions and variable dependencies explicit, helping readers understand complex models and reducing confusion between similar models.The authors present the framework as useful for teaching and communicating models to empiricists and theoreticians.Source: Discussion · p. 35
  • Graphical-model algorithms have well-studied counterparts to standard computational phylogenetics algorithms, offering a possible resource as phylogenetic models grow more complex.The authors describe this future benefit as a possibility, not a demonstrated outcome.Source: Discussion · p. 35
  • Graphical-model representations can connect phylogeneticists with applied statisticians through familiar model formulae such as y ∼ f(α, β).The authors suggest these formulae may make it easier for applied statisticians to contribute to phylogenetics.Source: Discussion · p. 35
  • Phylogenetic graphical models require capabilities that BUGS does not provide, including tree topologies as random variables and specialized computation for tree plates.The paper presents RevBayes as an independent implementation supporting PhyloGM objects and interactive model construction.Source: Software Implementation · pp. 36–37

Additional phylogenetic graphical models

The paper illustrates graphical-model representations for continuous traits, among-site rate variation, relaxed clocks, and the multispecies coalescent. These examples show how standard model components and tree plates can represent different phylogenetic dependencies.

A model of continuous character evolution

  • The Brownian-motion continuous-character model retains a graphical structure similar to the binary-character model, with a plate indicating replication across characters.The root state is drawn uniformly between min and max, and the variance parameter δ uniformly between x and y.Source: A model of continuous character… · p. 44, Figure S.1 · p. 44

A mixture model for rate variation among sites

  • In the GTR+Γ example, each site selects one of four equally probable rate categories, with rates set by gamma-distribution quantiles.The quantiles are 0.125, 0.375, 0.625, and 0.875; mixture categories can also be represented with indicator variables or marginalized.Source: A mixture model for rate variation… · p. 45, Figure S.2 · p. 45

A autocorrelated relaxed clock model

  • For an autocorrelated relaxed clock, log clock rates evolve by Brownian motion, and the tree plate emphasizes their dependence on tree topology.The rate for a branch is modeled as normal with mean equal to its parent-branch rate and variance t_iν.Source: A autocorrelated relaxed clock model · p. 46, Figure S.3 · p. 46

A graphical multispecies coalescent model

  • The multispecies coalescent graph places a birth-death prior on species-tree ages and applies independent coalescent processes to gene trees within species-tree branches.The graph emphasizes dependence between species-tree speciation times and gene-tree coalescent times; multiple genes can be represented on a plate.Source: A graphical multispecies coalescent… · p. 47, Figure S.4 · p. 47

RevBayes Examples

RevBayes examples specify both a simple non-phylogenetic Bernoulli model and a phylogenetic model of baculum presence across mammals. In the latter example, the estimated root and equilibrium frequencies are similar, and the data do not support the assumed difference between them.

RevBayes Examples

  • The non-phylogenetic example models presence or absence in five species with a Bernoulli distribution and estimates p.The Rev specification assigns p a beta prior and clamps the Bernoulli observations to the data.Source: A non-phylogenetic example · p. 48
  • The examples specify MCMC proposal moves that change model parameters; one run uses a 2,000-generation burn-in with a tuning interval of 100, followed by 200,000 generations.Source: A non-phylogenetic example · p. 49
  • The phylogenetic example models baculum presence or absence across 274 mammalian species with a continuous-time Markov model and a root frequency distinct from the equilibrium frequency.It uses a fixed tree and specifies priors for root and base frequencies.Source: A phylogenetic example · pp. 50–51
  • The estimated root frequency is 0.503 with HPD= {0.07, 0.91}, while the estimated equilibrium frequency is 0.48 with HPD= {0.38, 0.58}; the data do not support a difference.Source: A phylogenetic example · p. 52
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