Phylogenetic tree

A phylogenetic tree describes ancestral relationships. In a probabilistic graphical model on a rooted tree, latent variables at ancestral vertices are inferred from observations at leaves using belief propagation.

A phylogenetic epoch is an interval between successive branching or merging events during which the number of extant ancestral branches is constant. Its expected duration with $k$ branches is denoted $E_k$.

A representation of the branching history of species lineages. Individual gene trees can differ, so a well-supported tree for one gene is not automatically a complete species history.

A phylogenetic tree describing the ancestry of sampled gene copies or a genomic region. It need not match a species tree, because lineage sorting, duplication or introgression can produce a different history.

The Felsenstein pruning algorithm computes a site likelihood function by sum-product belief propagation from leaves to root. For a finite-state Markov kernel $K$, the subtree likelihood function obeys $L_v(a)={\prod }_{w\text{ child of }v}{\sum }_{b}K(a,b)L_w(b)$, with observed-state indicators at leaves. Summing against the root probability distribution gives the site likelihood function. An outward pass gives edge posterior probabilities and expected transition counts for the expectation-maximization algorithm.