Logarithmic bounds on the posterior divergence time of two sequences

When two initially identical binary sequences undergo independent site mutations at a constant rate, the proportion of site differences is often used to estimate the total time T that separates the two sequences. In this short note we study the posterior distribution of T when the prior distribution on T is exponential. We show that posterior estimates of T (for any data) cannot grow faster than the logarithm of the sequence length, and this rate is achieved for data generated at site saturation (i.e. in the limit as T → ∞ ). The problem is motivated by information-theoretic questions arising in molecular systematic biology, in which one wishes to use DNA sequences to estimate the divergence time between present-day species.