  The continuous-time Markov chain (CTMC) is the mathematical workhorse of
evolutionary biology. Learning CTMC model parameters using modern,
gradient-based methods requires the derivative of the matrix exponential
evaluated at the CTMC's infinitesimal generator (rate) matrix. Motivated by the
derivative's extreme computational complexity as a function of state space
cardinality, recent work demonstrates the surprising effectiveness of a naive,
first-order approximation for a host of problems in computational biology. In
response to this empirical success, we obtain rigorous deterministic and
probabilistic bounds for the error accrued by the naive approximation and
establish a "blessing of dimensionality" result that is universal for a large
class of rate matrices with random entries. Finally, we apply the first-order
approximation within surrogate-trajectory Hamiltonian Monte Carlo for the
analysis of the early spread of SARS-CoV-2 across 44 geographic regions that
comprise a state space of unprecedented dimensionality for unstructured
(flexible) CTMC models within evolutionary biology.
