Posterior bounds on divergence time of two sequences under dependent-site evolutionary models
Let $\x$ and $\y$ be two length $n$ DNA sequences, and suppose we would like to estimate the divergence time $T$. Under suitable conditions, a well-known simple but crude estimate of $T$ is the fraction of differing sites $\hat{p} := \text{d}_{\text{H}}(\x,\y)/n$ (the $p$-distance). We establish a posterior concentration bound on $T$, showing that the posterior distribution of $T$ concentrates within a logarithmic factor of $\hat{p}$ when $\text{d}_{\text{H}}(\x,\y)\log(n)/n = o(1)$. Our bounds hold under a large class of evolutionary models and prior distributions, including many standard models that incorporate site dependence. As a special case, we show that $T$ exceeds $\hat{p}$ with vanishingly small posterior probability as $n$ increases under models with constant mutation rates, complementing the result of Mihaescu and Steel (Appl Math Lett 23(9):975--979, 2010). Our approach is based on bounding sequence transition probabilities in various convergence regimes of the underlying evolutionary process. Our result is motivated by the problem of improving the efficiency of iterative optimization and sampling schemes for estimating divergence times in phylogenetic inference.