Analyzing the Accuracy of the Fitch Method for Reconstructing Ancestral States on Ultrametric Phylogenies

Recurrence formulas are presented for studying the accuracy of the Fitch method for reconstructing the ancestral states in a given phylogenetic tree. As their applications, we analyze the convergence of the accuracy of reconstructing the root state in a complete binary tree of $2^n$ as $n$ goes to infinity and also give a lower bound on the accuracy of reconstructing the root state in an ultrametric tree.

💡 Research Summary

The paper presents a rigorous probabilistic analysis of the classic Fitch parsimony algorithm for ancestral state reconstruction on phylogenetic trees. By modeling each edge as a stochastic process with a fixed mutation probability (or, in a continuous‑time setting, an exponential decay governed by a rate λ and branch length t), the authors derive recursive equations that give the probability that a particular character state survives in the Fitch set at any internal node. The key insight is that the Fitch set at a parent node can be expressed in terms of the probabilities of the two child sets: if the intersection of the child sets is non‑empty, the parent inherits the intersection; otherwise it inherits the union. Translating this deterministic rule into probability yields two cases: (1) the probability that a state s is present in both children (intersection) and (2) the probability that s appears in exactly one child (union). These cases lead to a compact recurrence:

 P_parent(s) = P_left(s)·P_right(s) if intersection non‑empty,
 P_parent(s) = P_left(s)·