by Thomas E. Currie and 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.