## Criticisms of the method

Phylogenetic applications of likelihood rely on two fundamental assumptions: that evolution is independent in different lineages as well as independent in different sites for a given tree, which are essential for the probability calculations on which the method is based (Felsenstein 1981b, 2004; Rodriguez et al. 1990; Swofford et al. 1996). Both assumptions are methodologically problematic because they are unrealistic and/or violated in the calculus (Siddall and Kluge 1997; Huelsenbeck and Nielsen 1999), but likelihoodists appeal to simulations to argue that the method is generally robust to violations of these assumptions (Yang 1994; Swofford et al. 1996). It is also assumed that the same stochastic process of substitution applies in all lineages (Felsenstein 1981b).

Many researchers advocate a model-based maximum likelihood approach (Felsenstein 1978, 1981b, 2004; Goldman 1990; Penny et al. 1992; Yang 1996; Huelsenbeck and Rannala 1997a; de Queiroz and Poe 2003). Others (Siddall and Kluge 1997; Kluge 2001; Goloboff 2003) cite operational and philosophical problems of likelihood methods and discourage its use in phylogenetic inference.

Likelihood methods rely on a specified model of sequence evolution to infer phylogenetic relationships. This is an inductive approach as the assumptions of the model are clearly deterministic to the result of the analysis. In phylogenetic-likelihood analyses, like all inductive approaches in science, all interpretations of the results come with the caveat "if the model is true, then '' Because likelihood is inductive estimation of phylogeny, particular caution must be taken in interpreting the results and to avoid circularity. We may know which of the models best fits the data according to ModelTest, but how can the validity of the model itself be independently tested? Testing of the validity of models, although it has been recognized as important (Goldman 1990), is rarely done in practice (Siddall and Kluge 1997). In a hypothetico-deductive framework, the assumptions of the method are background and not deterministic to the result. Only the observable data is considered, maximizing the explanatory power of the hypothesis and minimizing ad hoc hypotheses (i.e., hypotheses that confide in "nonobservables,'' such as long branches in a likelihood framework). It has been argued that many of the simple assumptions of the evolutionary models (i.e., the frequency of transitions versus transversions) constitute grounded knowledge about the process of molecular evolution and therefore is an acceptable background assumption. However, the maximum likelihood approach suggests that this is and has always been the case throughout all situations in the evolution of a group, which is a difficult assertion with respect to the historically contingent nature of the evolutionary process (Siddall and Kluge 1997).

Likelihood methods are based on frequency probability theory. Frequency probability is concerned with prediction of future events (Fisher 1922). The aim of phylogenetic systematics is to discover the unique evolutionary history of a group of organisms, or to elucidate its past (Kluge 1997). A species must be considered a historical entity (Kluge 1990), evolutionary transformations are unique and spatiotemporally restricted historical events (Siddall and Kluge 1997). Frequency-probability based methods of phylogenetic inference, such as maximum likelihood, apply frequency probability to a historical singularity, which is outside of the realm of future-predictive probability theory (Siddall and Kluge 1997). As noted above, likelihood methods assign all trees a nonzero probability, but in reality one tree has a probability of 1.0, and others have a probability ofzero. One must be cognizant that maximum likelihood inference of phylogenies is philosophically unsound because it employs frequency probability theory to estimate a nonprobabilistic phenomenon.

In spite of these numerous criticisms, likelihood approaches have become increasingly popular, particularly with molecular phylogeneticists. Recently, other methods incorporating likelihood logic have become popular in systematics, the most popular of which is Bayesian analysis (Huelsenbeck et al. 2002). Bayesian likelihood Bayesian methods calculate the posterior probabilities of phylo-genetic hypotheses (trees) using a version of Bayes' theorem in which the likelihood of the tree and the prior probability of the tree are considered (Huelsenbeck et al. 2002). Huelsenbeck et al. (2001, 2002) provide excellent recent review of the history and mechanics of Bayesian inference methods in phylogeny.

Reverend Thomas Bayes, living in the early half of the eighteenth century, was an English mathematician who was interested in the concept of using a priori knowledge to predict future events. His paper, "An Essay Towards Solving a Problem in the Doctrine of Chances,'' published 2 years after Bayes' death in 1761, introduced what would become known as Bayes' theorem (Eq. 10; Barnard and Bayes 1958).

The posterior probability, [P(H|D)], is the probability of the hypothesis given the observations, or data (D). Note that this is different from likelihood, which is the probability of the data given the hypothesis. However, the likelihood, P(D|H), is a parameter in the calculation of the posterior probability. P(H) is the prior probability of the hypothesis before the observation, data, or analysis, and reflects the original beliefs regarding the problem. The probability of the hypothesis is updated to take into account the observations, and Bayes' theorem describes the relationship between the prior and posterior probabilities. It was not until the latter half of the twentieth century that Bayes' ideas would be applied to the inference of phylogenies. Felsenstein (1968) briefly discussed Bayesian ideas as they could apply to phylogeny reconstruction in his Ph.D. thesis, but the statistical and computational framework with which to derive reliable approximations of posterior probabilities was not available at the time (Huelsenbeck et al. 2002).

Three independent groups introduced Bayesian methods similar to those currently in use in 1996 (Li 1996; Mau 1996; Rannala and Yang 1996). Bayesian methods to estimate ancestral character states have been developed (Pagel et al. 2004; Ronquist, 2004).

Huelsenbeck et al. (2001, 2002) provide overviews of the Bayesian methodology. Bayesian phylogenetic inference evaluates phylogenetic hypotheses with the posterior probabilities of trees. The posterior probability of each tree is calculated using the following Bayes-based equation (Eq. 11), where the tree topology (including branch lengths) is the hypothesis, and the data is typically molecular sequences of the terminal taxa in the analysis.

The likelihood parameter, P(Data j Tree), is calculated using the same general methodology and same models of molecular evolution described above for the maximum likelihood approach. However, one major difference between Bayesian and maximum likelihood methods is that Bayesian likelihood calculation not only involves summation over all possible combinations of model parameters and branch lengths but also includes a prior probability density distribution of these latter variables (Huelsenbeck et al. 2002). The prior probability of the tree, P (Tree), is usually considered to be equal for all trees a priori (Huelsenbeck et al.

2001). The use of equal priors implies that no particular topology is preferred over any other a priori and eliminates the sometimes difficult task of calculation of complex priors when hypotheses vary with respect to their preconceived probabilities. However, the prior for any given tree or set of trees can be set to reflect researcher experience, the results of previous analyses, or taxonomy (Huelsenbeck et al. 2002). The denominator, simplified here as P(Data), is a normalizing factor that involves summation over all trees (Huelsenbeck et al.

2002). The posterior probability, P(Tree j Data), can be viewed simply as the probability that the tree is "correct," given the data, the priors, and that the model of character change used in the likelihood calculation is correct (Huelsenbeck et al. 2002). There are several ways to present the results of a Bayesian analysis. The tree with the maximum posterior probability can be selected as the preferred phylogenetic hypothesis, this is also known as the MAP, maximum a posteriori, estimation of phylogeny (Huelsenbeck et al. 2002). Another method is to construct a 95% credibility consensus tree by starting with the MAP and consecutively adding the next most probable trees until the probabilities total 0.95 (Huelsenbeck et al. 2001, 2002). The method preferred by Huelsenbeck et al. (2002) is to present a summary of the results on the MAP or another consensus tree, as is typically done with bootstrap.

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