  The goal of phylodynamics, an area on the intersection of phylogenetics and
population genetics, is to reconstruct population size dynamics from genetic
data. Recently, a series of nonparametric Bayesian methods have been proposed
for such demographic reconstructions. These methods rely on prior
specifications based on Gaussian processes and proceed by approximating the
posterior distribution of population size trajectories via Markov chain Monte
Carlo (MCMC) methods. In this paper, we adapt an integrated nested Laplace
approximation (INLA), a recently proposed approximate Bayesian inference for
latent Gaussian models, to the estimation of population size trajectories. We
show that when a genealogy of sampled individuals can be reliably estimated
from genetic data, INLA enjoys high accuracy and can replace MCMC entirely. We
demonstrate significant computational efficiency over the state-of-the-art MCMC
methods. We illustrate INLA-based population size inference using simulations
and genealogies of hepatitis C and human influenza viruses.
