Bayes' theorem

A Bayesian approach to phylogeny reconstruction requires the definition of a parameter space $\Omega$ which contains the sets of all possible combined states $\phi=\{\tau_i,\nu_i,\theta\}$ where the symbol $\tau_i$ labels the $i^{th}$ possible tree topology, $\nu_i$ are the branch lengths associated with this topology and $\theta$ is a set of allowed parameters for our evolutionary model (e.g., rate ratios $\alpha_{ij}$, nucleotide or base-pair frequencies $\pi_i$, gamma distribution parameter $\alpha$, ...). According to Bayes' theorem, we can calculate the posterior probability of the combined state $\phi$ given sequence data $X$,

$$p(\phi\vert X) = \frac{P(X\vert\phi) p(\phi)}{\sum_{i=1}^{N_s}\int d\nu_i \int d\theta P(X\vert\phi)p(\phi)}$$ (5)

where $N_s$ is the number of possible tree topologies for a data set containing $s$ species, $P(X\vert\phi)$ is the likelihood of the data and $p(\phi)$ is the prior probability density associated with state $\theta$.