We develop a Bayesian inference method for diffusions observed discretely and with noise, which is free of discretization bias. Unlike existing unbiased inference methods, our method does not rely on exact simulation techniques. Instead, our method uses standard time-discretized approximations of diffusions, such as the Euler--Maruyama scheme. Our approach is based on particle marginal
Metropolis--Hastings, a particle filter, randomized multilevel Monte Carlo, and an importance sampling type correction of approximate Markov chain Monte Carlo. The resulting estimator leads to inference without a bias from the time-discretization as the number of Markov chain iterations increases. We give convergence results and recommend allocations for algorithm inputs. Our method
admits a straightforward parallelization and can be computationally efficient. The user-friendly approach is illustrated on three examples, where the underlying diffusion is an Ornstein--Uhlenbeck process, a geometric Brownian motion, and a 2d nonreversible Langevin equation.
Chada, N. K., Franks, J., Jasra, A., Law, K. J., & Vihola, M. (2021). Unbiased Inference for Discretely Observed Hidden Markov Model Diffusions. SIAM/ASA Journal on Uncertainty Quantification, 9(2), 763–787. doi:10.1137/20m131549x
The work of the first and third authors was supported by KAUST baseline funding. The work of the second, third, fourth, and fifth authors was supported by the Academy of Finland (grants 274740, 312605, and 315619) and by the Institute for Mathematical Sciences, Singapore (2018 programme ``Bayesian Computation for High-Dimensional Statistical Models""). The work of the second and fourth authors was also supported by The Alan Turing Institute. The work of the third author was also supported by the Singapore Ministry of Education (R-155-000-161-112). The work of the fourth author was also supported by the University of Manchester (School of
Mathematics). This research made use of the Rocket High Performance Computing service at Newcastle University.