# Unbiased inference for discretely observed hidden Markov model diffusions

arXiv: https://arxiv.org/abs/1807.10259 · HTML: https://mlsanity.com/paper/1807.10259/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jul 2018.

## Built on by (most cited first, 2 of 2)

- [Unbiased least squares regression via averaged stochastic gradient descent](https://mlsanity.com/paper/2406.18623/) (2024)
- [Theoretical and Practical Advances in Hierarchical Bayesian Deep Learning Methodology](https://mlsanity.com/paper/2610.19074/) (2026)

## Abstract (arXiv)

We develop a Bayesian inference method for diffusions observed discretely and with noise, which is free of discretisation bias. Unlike existing unbiased inference methods, our method does not rely on exact simulation techniques. Instead, our method uses standard time-discretised approximations of diffusions, such as the Euler--Maruyama scheme. Our approach is based on particle marginal Metropolis--Hastings, a particle filter, randomised multilevel Monte Carlo, and importance sampling type correction of approximate Markov chain Monte Carlo. The resulting estimator leads to inference without a bias from the time-discretisation as the number of Markov chain iterations increases. We give convergence results and recommend allocations for algorithm inputs. Our method admits a straightforward parallelisation, 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 non-reversible Langevin equation.