Particle-MALA and Particle-mGRAD: Gradient-based MCMC methods for high-dimensional state-space models

State-of-the-art methods for Bayesian inference in state-space models are (a) conditional sequential Monte Carlo (CSMC) algorithms; (b) sophisticated 'classical' MCMC algorithms like MALA, or mGRAD from Titsias and Papaspiliopoulos (2018, arXiv:1610.09641v3 [stat.ML]). The former propose N particles at each time step to exploit the model's 'decorrelation-over-time' property and thus scale favourably with the time horizon, T , but break down if the dimension of the latent states, D, is large. The latter leverage gradient-/prior-informed local proposals to scale favourably with D but exhibit sub-optimal scalability with T due to a lack of model-structure exploitation. We introduce methods which combine the strengths of both approaches. The first, Particle-MALA, spreads N particles locally around the current state using gradient information, thus extending MALA to T > 1 time steps and N > 1 proposals. The second, Particle-mGRAD, additionally incorporates (conditionally) Gaussian prior dynamics into the proposal, thus extending the mGRAD algorithm to T > 1 time steps and N > 1 proposals. We prove that Particle-mGRAD interpolates between CSMC and Particle-MALA, resolving the 'tuning problem' of choosing between CSMC (superior for highly informative prior dynamics) and Particle-MALA (superior for weakly informative prior dynamics). We similarly extend other 'classical' MCMC approaches like auxiliary MALA, aGRAD, and preconditioned Crank-Nicolson-Langevin (PCNL) to T > 1 time steps and N > 1 proposals. In experiments, for both highly and weakly informative prior dynamics, our methods substantially improve upon both CSMC and sophisticated 'classical' MCMC approaches.