Figure 3.

The evolution of a single binary character represented as a phylogenetic graphical model. a) The phylogenetic relationships of the five mammalian species. The observed state of the character (1: presence or 0: absence of the baculum) is given for each species. Other states at the internal nodes represent the unknown ancestral state. The branches of the tree (1,…,8) are labeled and assigned a fixed length (l₁,…,l₈). b) The corresponding graphical model, in which the species tree topology is still evident. We represent the state for each node with generic notation: S₁ is the presence/absence state for node 1. The clamped nodes, in grey, indicate observed states, whereas unobserved states for ancestral species are in white. Constant nodes indicate fixed/known branch lengths. Under this model, the state for the root of the tree (S₉) is drawn from a Bernoulli distribution with probability p. A Beta prior is assigned to the parameter of the Bernoulli distribution so that p∼Beta(α,β), where the parameters of the Beta distribution are constant nodes and assigned fixed values. The states of the nodes descended from the root of the tree (S₁,…,S₈) are dependent on the equilibrium frequency parameter (θ) and their respective branch lengths (constant nodes l₁,…,l₈). A second Beta distribution is applied as a prior on the parameter θ, where θ∼Beta(x,y).