Particle Metropolis-adjusted Langevin algorithms

In this paper we propose a new sampling scheme based on Langevin dynamics that is applicable in pseudo-marginal and particle Markov chain Monte Carlo algorithms. We investigate the algorithm's theoretical properties under standard asymptotics, which correspond to an increasing dimension $n$ of the parameters. Our results show that the behaviour of the algorithm depends crucially on how accurately one can estimate the gradient of the log target density. If the error in the estimate of the gradient is not sufficiently controlled as the dimension increases, then asymptotically there will be no advantage over the simpler random-walk algorithm. However, if the error is sufficiently well-behaved, then the optimal scaling of this algorithm will be $O(n^{-1/6})$, compared to $O(n^{-1/2})$ for the random walk. Our theory also gives guidelines on how to tune the number of Monte Carlo samples in the likelihood estimate and the proposal step-size.