Fig. 1

A preview of the core components of the algorithm to give intuition; concepts will be introduced in the text. a The subsplit DAG, which encodes a collection of phylogenetic tree topologies on leaves \(x_1, \ldots , x_N\). One such topology is partially shown in black, with alternate topologies indicated with gray lines. The nodes of this DAG are uniquely associated with “subsplits” that give the bipartition of taxa below them (subsplits corresponding to alternate topologies are marked with \(\cdot _{\text {alt}}\)). For example, the subsplit t is \((\{x_{i+1}, \ldots , x_j\},\{x_{j+1}, \ldots , x_l\})\). Edges go between compatible subsplits, such as between t and s, and are directed towards the leaves. b Overview of method: given an edge of the DAG we integrate (\(\int\)) out all of the topologies in the DAG that contain the DAG edge \(t \rightarrow s\). Branch lengths \(\theta\) are associated to DAG edges. We perform efficient inference using “rootward” \(\textbf{r}\) and “leafward” \({\textbf{p}}\) partial likelihood vectors marginalized over unknown structure of trees encoded in the DAG