Keeping Yourself Updated: Bayesian Approaches in Phylogenetic Comparative Methods with a Focus on Markov Chain Models of Discrete Character Evolution

  • Thomas E. CurrieEmail author
  • Andrew Meade


Bayesian inference involves altering our beliefs about the probability of events occurring as we gain more information. It is a sensible and intuitive approach that forms the basis of the kinds of decisions we make in everyday life. In this chapter, we examine how phylogenetic comparative methods are performed within a Bayesian framework, introducing some of the main concepts involved in Bayesian statistics, such as prior and posterior distributions. Many traits of biological and evolutionary interest can be modelled as being categorical, or discretely distributed, and here, we discuss approaches to investigating the evolution of such characters over phylogenetic trees. We focus on Markov chain models of discrete character evolution and how these models can be assessed using maximum-likelihood and Markov Chain Monte Carlo techniques of parameter estimation. We demonstrate how this can be used to test functional hypotheses by examining the correlated evolution of different traits, illustrated with examples of sexual selection in primates and cichlid fish. We show how the order of trait evolution can be determined (potentially providing a stronger test of causal hypotheses) and how competing hypotheses can be assessed using Bayes factors. Attractive features of these Bayesian methods are their ability to incorporate uncertainty about the phylogenetic relationships between species and their representation of results as probability distributions rather than point estimates. We argue that Bayesian methods provide a more realistic way of assessing evidence and ultimately a more intellectually satisfying approach to investigating the diversity of life.



Bayes factors

Bayes factors are a way of testing between different hypotheses in a Bayesian framework. They are calculated as ratios of the marginal likelihoods of different models. The larger the ratio the more support there is for one model over another. The interpretation of Bayes factors is somewhat arbitrary, but rules of thumb exist to use these values to assess the strength of evidence in favour of one hypothesis over another.


A hyperprior is a prior distribution on the hyperparameter of a prior distribution. In other words instead of the parameters of the distribution being given fixed values, they themselves are drawn from prior distributions. In the program BayesTraits (which is used in the OPM), the hyperparameters of specified distributions are drawn from uniform distributions. For example, a gamma distribution could have its shape and scale parameters drawn from a uniform distribution ranging from 0 to 10. In comparative analyses, we do not always possess relevant biological information that could inform us about what form and values the priors should take therefore hyperpriors are attractive because they allow us to be less restrictive about the values of a given prior distribution.

Marginal Likelihood

The marginal likelihood of a model is its likelihood scaled by the prior probabilities and integrated over all values of the parameters. In the context of phylogenetic comparative analyses this may also involve integrating over all the trees in the sample.

Markov Chain Monte Carlo (MCMC)

MCMC is a statistical procedure used in Bayesian analyses to search parameter space and sample values in proportion to their posterior probability in order to arrive at an estimate of the posterior distributions of a model and its parameter values. A number of different criteria can be implemented to govern the way an MCMC searches and samples the posterior distribution. With the MetropolisHasting algorithm, parameter values that increase the likelihood are always accepted, while those that lead to a decrease are accepted only with a certain probability. The Gibbs sampler always accepts proposed values but works by drawing new values from the conditional distributions of the parameters (i.e. the distribution of a parameter given the value of other parameters).

Maximum likelihood

In a maximum-likelihood we search for the values of the parameters of a statistical model that give the largest possible value of the likelihood function.

Prior probability and Priors/Prior distribution

In Bayesian statistics, we need to specify our initial belief about the probability of a hypothesis, given the information available at the time. This belief then gets updated when we gain more information (i.e. this is our belief prior to the assessment of new information). In the context of a comparative analysis, we are assessing the parameters of a statistical model, and before running the analysis and examining the data, we have to specify a prior probability distribution of the values these parameters should take given our current understanding. The chapter by Currie and Meade provides some examples of common prior distributions that are used in comparative analyses. See also Hyperprior.

Posterior probability and Posteriors/Posterior distribution

In Bayesian statistics, the posterior probability refers to our belief in a hypothesis after (i.e. posterior to) assessing new information. In the context of a comparative analysis, the results of our analysis give us the posterior probability distribution of values of the parameters of a statistical model. See also Markov Chain Monte Carlo (MCMC).


  1. Arnold C, Matthews LJ, Nunn CL (2010) The 10kTrees website: a new online resource for primate phylogeny. Evol Anthropol 19:114–118CrossRefGoogle Scholar
  2. Beaulieu JM, Jhwueng DC, Boettiger C, O’Meara BC (2012) Modeling stabilizing selection: expanding the Ornstein–Uhlenbeck model of adaptive evolution. Evolution 66(8):2369–2383. doi: 10.1111/j.1558-5646.2012.01619.x CrossRefPubMedPubMedCentralGoogle Scholar
  3. Bollback JP (2006) SIMMAP: stochastic character mapping of discrete traits on phylogenies. BMC Bioinformatics 7(1):88. doi  10.1186/1471-2105-7-88 CrossRefGoogle Scholar
  4. Burnham KP, Anderson DR (2002) Model selection and multi-model inference: a practical information-theoretic approach. Springer, New YorkGoogle Scholar
  5. Cardinal S, Straka J, Danforth BN (2009) Comprehensive phylogeny of apid bees reveals the evolutionary origins and antiquity of cleptoparasitism. In: Proceedings of the National Academy of Sciences. doi: 10.1073/pnas.1006299107 CrossRefGoogle Scholar
  6. Currie TE, Greenhill SJ, Gray RD, Hasegawa T, Mace R (2010) Rise and fall of political complexity in island south-east Asia and the Pacific. Nature 467(7317):801–804CrossRefGoogle Scholar
  7. Davies NB, Krebs JR, West SA (2012) An introduction to behavioural ecology. Wiley, New JerseyGoogle Scholar
  8. de Villemereuil P, Wells J, Edwards R, Blomberg S (2012) Bayesian models for comparative analysis integrating phylogenetic uncertainty. BMC Evol Biol 12(1):102CrossRefGoogle Scholar
  9. Domb LG, Pagel M (2001) Sexual swellings advertise female quality in wild baboons. Nature 410(6825):204–206CrossRefGoogle Scholar
  10. Dunn M, Greenhill SJ, Levinson SC, Gray RD (2011) Evolved structure of language shows lineage-specific trends in word-order universals. Nature 473(7345):79–82CrossRefGoogle Scholar
  11. Felsenstein J (2005) Using the quantitative genetic threshold model for inferences between and within species. Philos Trans Roy Soc B Biol Sci 360(1459):1427–1434. doi: 10.1098/rstb.2005.1669 CrossRefGoogle Scholar
  12. Felsenstein J (2012) A comparative method for both discrete and continuous characters using the threshold model. Am Nat 179(2):145–156. doi: 10.1086/663681 CrossRefGoogle Scholar
  13. Fitzpatrick JL, Montgomerie R, Desjardins JK, Stiver KA, Kolm N, Balshine S (2009) Female promiscuity promotes the evolution of faster sperm in cichlid fishes. Proc Natl Acad Sci 106(4):1128–1132. doi: 10.1073/pnas.0809990106 CrossRefPubMedGoogle Scholar
  14. Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4):711–732CrossRefGoogle Scholar
  15. Griffin RH, Matthews LJ, Nunn CL (2012) Evolutionary disequilibrium and activity period in primates: a bayesian phylogenetic approach. Am J Phys Anthropol 147(3):409–416. doi: 10.1002/ajpa.22008 CrossRefPubMedGoogle Scholar
  16. Hadfield JD (2010) MCMC methods for multi-response generalized linear mixed models: the mcmcglmm R package. J Stat Softw 33(2):122CrossRefGoogle Scholar
  17. Hibbett DS (2004) Trends in morphological evolution in homobasidiomycetes inferred using maximum likelihood: a comparison of binary and multistate approaches. Syst Biol 53(6):889–903CrossRefGoogle Scholar
  18. Holden C, Mace R (1997) Phylogenetic analysis of the evolution of lactose digestion in adults. Hum Biol 69(5):605–628PubMedGoogle Scholar
  19. Huelsenbeck JP, Nielsen R, Bollback JP (2003) Stochastic mapping of morphological characters. Syst Biol 52(2):131–158. doi: 10.1080/10635150390192780 CrossRefPubMedGoogle Scholar
  20. Huelsenbeck JP, Rannala B (2003) Detecting correlation between characters in a comparative analysis with uncertain phylogeny. Evolution 57(6):1237–1247CrossRefGoogle Scholar
  21. Huelsenbeck JP, Rannala B, Masly JP (2000) Accommodating phylogenetic uncertainty in evolutionary studies. Science 288(5475):2349–2350CrossRefGoogle Scholar
  22. Ives AR, Garland T (2010) Phylogenetic logistic regression for binary dependent variables. Syst Biol 59(1):9–26. doi: 10.1093/sysbio/syp074 CrossRefPubMedPubMedCentralGoogle Scholar
  23. Jordan FM, Gray RD, Greenhill SJ, Mace R (2009) Matrilocal residence is ancestral in Austronesian societies. Proc Roy Soc B Biol Sci 276(1664):1957–1964. doi: 10.1098/rspb.2009.0088 CrossRefGoogle Scholar
  24. Kass RE, Raftery AE (1995) Bayes Factors. J Am Stat Assoc 90(430):773–795CrossRefGoogle Scholar
  25. Link WA, Barker RJ (2009) Bayesian inference: with ecological applications. Elsevier Science, New JerseyGoogle Scholar
  26. Maddison WP (1990) A method for testing the correlated evolution of 2 binary characters—are gains or losses concentrated on certain branches of a phylogenetic tree. Evolution 44(3):539–557CrossRefGoogle Scholar
  27. Maddison WP, Maddison DR (2009) Mesquite: a modular system for evolutionary analysis. 2.71 ednGoogle Scholar
  28. O’Meara BC, Ane C, Sanderson MJ, Wainwright PC (2006) Testing for different rates of continuous trait evolution using likelihood. Evolution 60(5):922–933CrossRefGoogle Scholar
  29. Organ CL, Janes DE, Meade A, Pagel M (2009) Genotypic sex determination enabled adaptive radiations of extinct marine reptiles. Nature 461(7262):389–392CrossRefGoogle Scholar
  30. Pagel M (1994a) Detecting correlated evolution on phylogenies: a general-method for the comparative-analysis of discrete characters. Proc R Soc Lond Ser B Biol Sci 255(1342):37–45CrossRefGoogle Scholar
  31. Pagel M (1994b) Evolution of conspicuous estrous advertisement in old-world monkeys. Anim Behav 47(6):1333–1341CrossRefGoogle Scholar
  32. Pagel M (1999) Inferring the historical patterns of biological evolution. Nature 401(6756):877–884CrossRefGoogle Scholar
  33. Pagel M, Meade A (2005) Bayesian estimation of correlated evolution across cultures: a case study of marriage systems and wealth transfer at marriage. In: Mace R, Holden CJ, Shennan S (eds) Left Coast Press. Walnut Creek, CaliforniaGoogle Scholar
  34. Pagel M, Meade A (2006) Bayesian analysis of correlated evolution of discrete characters by reversible-jump Markov chain Monte Carlo. Am Nat 167(6):808–825PubMedPubMedCentralGoogle Scholar
  35. Pagel M, Meade A, Barker D (2004) Bayesian estimation of ancestral character states on phylogenies. Syst Biol 53(5):673–684CrossRefGoogle Scholar
  36. Pagel M, Venditti C, Meade A (2006) Large punctuational contribution of speciation to evolutionary divergence at the molecular level. Science 314(5796):119–121. doi: 10.1126/science.1129647 CrossRefPubMedGoogle Scholar
  37. Penny D, McComish BJ, Charleston MA, Hendy MD (2001) Mathematical elegance with biochemical realism: the covarion model of molecular evolution. J Mol Evol 53(6):711–723. doi: 10.1007/s002390010258 CrossRefPubMedPubMedCentralGoogle Scholar
  38. Posada D (2009) Selecting models of evolution. In: Lemey P, Salemi M, Vandamme AM (eds) The phylogenetic handbook: a practical approach to phylogenetic analysis and hypothesis testing. Cambridge University Press, Cambridge, pp 345–361CrossRefGoogle Scholar
  39. Ronquist F, van der Mark P, Huelsenbeck JP (2009) Bayesian phylogenetic analysis using MrBayes. In: Lemey P, Salemi M, Vandamme AM (eds) The phylogenetic handbook: a practical approach to phylogenetic analysis and hypothesis testing. Cambridge University Press, CambridgeGoogle Scholar
  40. Sanderson MJ (1993) Reversibility in evolution: a maximum-likelihood approach to character gain loss bias in phylogenies. Evolution 47(1):236–252CrossRefGoogle Scholar
  41. Shultz S, Opie C, Atkinson QD (2011) Stepwise evolution of stable sociality in primates. Nature 479(7372):219–222CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Centre for Ecology and Conservation, Biosciences, College of Life and Environmental SciencesUniversity of ExeterPenryn, CornwallUK
  2. 2.School of Biological SciencesUniversity of ReadingReading, BerkshireUK

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