  In this paper we consider Bayesian parameter inference for partially observed
fractional Brownian motion (fBM) models. The approach we follow is to
time-discretize the hidden process and then to design Markov chain Monte Carlo
(MCMC) algorithms to sample from the posterior density on the parameters given
data. We rely on a novel representation of the time discretization, which seeks
to sample from an approximation of the posterior and then corrects via
importance sampling; the approximation reduces the time (in terms of total
observation time T) by O(T). This method is extended by using a multilevel MCMC
method which can reduce the computational cost to achieve a given mean square
error (MSE) versus using a single time discretization. Our methods are
illustrated on simulated and real data.
