Bayesian inference is a statistical method based on Bayes\u2019 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 \u201cconditional probability\u201d, the probability of an outcome given<\/em> another outcome (Casella 2008; Puga et al. 2015b). At the core of Bayesian inference is Bayes\u2019 theorem (Puga et al. 2015a), in which the probability of a model M<\/em> given the data D<\/em> is described by P<\/em>(M<\/em>\u01c0D<\/em>), and it is calculated as follows:<\/p>\n Here, P<\/em>(D<\/em>\u01c0M<\/em>) is referred to as the likelihood<\/em>, and it describes the compatibility of the data, given a model (specifically, it is the probability of the model M<\/em> producing the data D)<\/em>. The P<\/em>(M<\/em>) is the probability of the model M<\/em> before the data D<\/em> are observed, also known as the prior probability<\/em> or simply a prior<\/em>. 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<\/em>(M<\/em>\u01c0D<\/em>). The P<\/em>(M<\/em>\u01c0D<\/em>) is referred to as the posterior probability<\/em>. Finally, P<\/em>(D<\/em>) represents the probability of the data. Critically, when posterior probabilities are calculated using the same data, P<\/em>(D<\/em>) 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<\/em>(M<\/em>\u01c0D<\/em>) and is often ignored:<\/p>\n 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 M1<\/sub><\/em> over model M2<\/sub><\/em>) to the posterior odds of the hypotheses (Kass and Raftery 1995). The Bayes factor K<\/em> is thus the ratio of the two marginal likelihoods of the models:<\/p>\n Conveniently, Kass and Raftery (1995) provide a scale to discriminate between models based on the value of the ratio K<\/em>.\u00a0 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\u00ad-values<\/em>). It is also important to remember when calculating Bayes factors that most coalescent or phylogenetic software packages report probabilities and likelihoods in loge<\/sub> units (this is done because likelihoods of phylogenies and genealogies can be exceeding small).<\/p>\n <\/p>\n<\/p><\/div>\n<\/section>\n","protected":false},"excerpt":{"rendered":" Bayesian inference is a statistical method based on Bayes\u2019 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 \u201cconditional probability\u201d, the probability of an outcome given<\/em> another outcome (Casella 2008; Puga et al. 2015b). At the core of Bayesian inference is Bayes\u2019 theorem (Puga et al. 2015a), in which the probability of a model M<\/em> given the data D<\/em> is described by P<\/em>(M<\/em>\u01c0D<\/em>), and it is calculated as follows:<\/p>\n![\"\"](\"https:\/\/wpcdn.web.wsu.edu\/wp-hub\/uploads\/sites\/1198\/2018\/10\/equation-1-396x128.png\")
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