  It has been widely documented that the sampling and resampling steps in
particle filters cannot be differentiated. The {\itshape reparameterisation
trick} was introduced to allow the sampling step to be reformulated into a
differentiable function. We extend the {\itshape reparameterisation trick} to
include the stochastic input to resampling therefore limiting the
discontinuities in the gradient calculation after this step. Knowing the
gradients of the prior and likelihood allows us to run particle Markov Chain
Monte Carlo (p-MCMC) and use the No-U-Turn Sampler (NUTS) as the proposal when
estimating parameters.
  We compare the Metropolis-adjusted Langevin algorithm (MALA), Hamiltonian
Monte Carlo with different number of steps and NUTS. We consider two
state-space models and show that NUTS improves the mixing of the Markov chain
and can produce more accurate results in less computational time.
