GoKawiilFor brevity, we summarize the essentials of Delphy’s operation here; full details are provided in the Supplementary Information.
EMATs
Delphy represents trees internally as a collection of explicitly timed nodes and a reference sequence. Each non-root node points to an earlier parent, and each inner node to two later children. Every instant in the tree has an associated sequence, encoded as successive differences from a reference sequence consisting of L states, each in {A,C,G,T}. The root node stores mutations (a,ℓ,b) recording a difference at site ℓ between the reference state a and the root sequence state b; we refer to these mutations as being above the root node. Every other node is annotated with a sequence of mutations from its parent to itself. Each mutation (a,ℓ,b,t) records a change in state at time t from a to b. Thus, the sequence at point x on the tree is obtained starting with the reference sequence, applying the mutations above the root node to obtain the root sequence, then successively applying in order all mutations on the unique path from the root node to x.
Nodes are also annotated with missations, tuples (a, ℓ) recording that for all tips downstream of this node, but not its parent, the state of site ℓ is unknown (missing, ‘N’ in the input); at the parent, the state is a. The tree topology, mutational history and tip sequences jointly completely determine all missations. Missations are encoded using two complementary structures: an ordered sequence of disjoint, non-consecutive half-open intervals [ℓ start , ℓ end ); and a map of sites ℓ to states a whenever the reference state is not a. This representation reflects that missing data typically appears in a few long gaps, and that the site-to-state map is sparse, as the state at site ℓ is typically the root state when root-to-tip times are small compared with mutation rates (assuming a reference sequence matching a representative root sequence).
To support uncertain tip dates, each tip has a minimum and maximum time, which coincide when there is no uncertainty.
EMATs have evident consistency requirements for node times, mutation times and identities and missation states (Supplementary Information).
Posterior distribution
Delphy samples trees \({\mathcal{T}}\) and associated model parameters θ using MCMC according to the following (unnormalized) posterior distribution:
$$P({\mathcal{T}},\theta )\propto L({\mathcal{T}}\,)\times {\mathcal{G}}({\mathcal{T}}|\theta )\times {\pi }_{\mathrm{anc}}({\mathcal{T}}|\theta )\times {\pi }_{\theta }(\theta )$$ (1)
The factors are as follows:
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