  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.
