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