Empirical and Hierarchical Bayesian Estimation of Ancestral States

—Several methods have been proposed to infer the states at the ancestral nodes on a phylogeny. These methods assume a specic tree and set of branch lengths when estimating the ancestral character state. Inferences of the ancestral states, then, are conditioned on the tree and branch lengths being true. We develop a hierarchical Bayes method for inferring the ancestral states on a tree. The method integrates over uncertainty in the tree, branch lengths, and substitution model parameters by using Markov chain Monte Carlo. We compare the hierarchicalBayes inferences of ancestral states with inferences of ancestral states made under the assumption that a specic tree is correct. We nd that the methods are correlated, but that accommodating uncertainty in parameters of the phylogenetic model can make inferences of ancestral states even more uncertain than they would be in an empirical Bayes analysis. [Ancestral state reconstruction; Bayesian estimation; empirical Bayes; hierarchical Bayes. ] The reconstruction of ancestral states on a phylogeny remains an important endeavor in evolutionary biology. The comparative method, for example, looks for evidence of correlated change in two or more characters; in the course of such analyses, the ancestral states are often reconstructed on a tree (Harvey and Pagel, 1991). Other studies examine the properties of ancient molecules (Malcolm et al., 1990; Stackhouse et al., 1990; Adey et al., 1994; Jermann et al., 1995). The amino acid sequence of the ancient protein is estimated and then synthesized in the laboratory. The properties of the ancient protein can then be measured in vivo or in vitro with the goal of demonstrating a change in the function of the protein. Similar studies have been performed for morphological or behavioral traits. For example, Ryan and Rand (1995) reconstructed the mating calls of the hypothetical ancestors of a group of frogs and then examined the response of the extant females to the ancient calls. The inferences made in such studies depend on the reliability of the ancestral state reconstructions. Several methods have been proposed to reconstruct the states present in the hypothetical ancestors on a phylogenetic tree. The parsimony method, probably the most frequently used method to infer ancestral states, nds the combination of ancestral states at an interior node of a phylogeny that minimizes the number of changes over the whole tree. The result of a parsimony analysis of ancestral states is either to choose one state as the best reconstruction or, less frequently, to present multiple reconstructions when several reconstructionsgive the same parsimony tree length (in which case the reconstruction is said to be ambiguous). More recently, ancestral states have been reconstructed for DNA, amino acid, and morphological (two-state) data by the use of stochastic models (Schluter, 1995; Yang et al., 1995; Schluter et al., 1997; Pagel, 1999). Two different methods have been used to infer the ancestral states by using stochastic models. The maximum likelihood method nds the character state at an internal node on the tree that maximizes the probability of observing the data. For Bayesian inference, on the other hand, the goal is to calculate what is called the posterior probability that an ancestral node on a tree has a particular state, given the observations at the tips of the tree. The probability that a character takes a particular state at some interior node on a phylogenetic tree depends on the topology of the phylogenetic tree, the lengths of the branches on the phylogeny, and the parameters of the substitution model (such as the transition/transversion rate bias). The typical approach is to use the maximum likelihood estimates of these parameters when calculating the posterior probability of a site. Such an analysis is referred to as an empirical Bayes analysis. Schultz and Churchill (1999) have reviewed Bayesian approaches to reconstructing ancestral states for morphological characters. In this study, we examine how sensitive the empirical Bayesian estimates of ancestral states are to uncertainty in the tree, branch lengths, and substitution parameters. We propose a method that integrates over uncertainty in these parameters; such an 351 D ow naded rom http/academ ic.p.com /sysbio/articlact/50/3/351/1661227 by gest on 11 M arch 2019 352 SYSTEMATIC BIOLOGY VOL. 50 analysis is referred to as an hierarchical Bayes analysis. The advantage of a hierarchical Bayes analysis is that inferences about the state of an ancestral character are not conditioned on any single tree or set of parameter values. We approximate the posterior probability of an ancestral state assignment using Markov chain Monte Carlo (MCMC). We nd that it is important to consider uncertainty in model parameters and phylogeny when reconstructing ancestral states.