it has been recognised that the gradient and Hessian can be used to construct efficient proposal distributions. In the Metropolis adjusted Langevin algorithm (MALA) [@RobertsStramer2003], a drift term is added to the proposal in the direction of the gradient, which intuitively guides the Markov chain to regions of high posterior probability. In the manifold MALA (mMALA) [@GirolamiCalderhead2011], the Hessian (or some other appropriate metric tensor) is also included to scale the proposal to take the curvature of the log-posterior into account. Drawing parallels with the optimisation literature, mMALA shares some properties with Newton-type optimisation algorithms (where MALA is more similar to a steepest ascent method). In particular, scaling the proposal with the Hessian can considerably simplify the tedious tuning of the method since it removes the need for running costly pilot runs, which are commonly used to tune the covariance matrices of the random walk MH and the MALA.

In our problem, i.e. for inference in a nonlinear SSM , the gradient and Hessian cannot be computed analytically. However, in analogue with the intractable likelihood, these quantities can be estimated using SMC algorithms, see e.g. @PoyiadjisDoucetSingh2011 [@DoucetJacobRubenthaler2013]. This provides us with the tools necessary to construct PMH algorithms in the 