^{Fernando Villanea}

_{Bayesian Inference}

Bayesian inference is a statistical method based on Bayes’ theorem, in which the probability of a hypothesis is updated based on prior evidence and a model created to explain the data (Konigsberg and Frankenberg 2013). In Bayesian inference probability is treated as “conditional probability”, the probability of an outcome *given* another outcome (Casella 2008; Puga et al. 2015b). At the core of Bayesian inference is Bayes’ theorem (Puga et al. 2015a), in which the probability of a model *M* given the data *D* is described by *P*(*M*ǀ*D*), and it is calculated as follows:

Here, *P*(*D*ǀ*M*) is referred to as the *likelihood*, and it describes the compatibility of the data, given a model (specifically, it is the probability of the model *M* producing the data *D)*. The *P*(*M*) is the probability of the model *M* before the data *D* are observed, also known as the *prior probability* or simply a *prior*. A prior represents our degree of belief in the values that a parameter can take, and it modifies the likelihood to produce the probability of a model given the data *P*(*M*ǀ*D*). The *P*(*M*ǀ*D*) is referred to as the *posterior probability*. Finally, *P*(*D*) represents the probability of the data. Critically, when posterior probabilities are calculated using the same data, *P*(*D*) takes the same value in all independent calculations (as the empirical data are the same for all), it is therefore a fixed scalar of *P*(*M*ǀ*D*) and is often ignored:

Model testing in Bayesian frameworks are relatively straightforward and is usually performed using Bayes factors. A Bayes factor is the ratio of the prior odds of two hypotheses (i.e., the odds of model *M _{1}* over model

*M*) to the posterior odds of the hypotheses (Kass and Raftery 1995). The Bayes factor

_{2}*K*is thus the ratio of the two marginal likelihoods of the models:

Conveniently, Kass and Raftery (1995) provide a scale to discriminate between models based on the value of the ratio *K*. Notably, though this scale has assumed some authority in the field of Bayesian inference, it is itself a suggestion when interpreting the importance of Bayes factor values (as are schemes regarding the significance of *p-values*). It is also important to remember when calculating Bayes factors that most coalescent or phylogenetic software packages report probabilities and likelihoods in log_{e} units (this is done because likelihoods of phylogenies and genealogies can be exceeding small).