Paul Horridge

Joshua Murphy, Conor Rosato, Andrew Millard et al.

When performing Bayesian inference using Sequential Monte Carlo (SMC) methods, two considerations arise: the accuracy of the posterior approximation and computational efficiency. To address computational demands, Sequential Monte Carlo Squared (SMC$^2$) is well-suited for high-performance computing (HPC) environments. The design of the proposal distribution within SMC$^2$ can improve accuracy and exploration of the posterior as poor proposals may lead to high variance in importance weights and particle degeneracy. The Metropolis-Adjusted Langevin Algorithm (MALA) uses gradient information so that particles preferentially explore regions of higher probability. In this paper, we extend this idea by incorporating second-order information, specifically the Hessian of the log-target. While second-order proposals have been explored previously in particle Markov Chain Monte Carlo (p-MCMC) methods, we are the first to introduce them within the SMC$^2$ framework. Second-order proposals not only use the gradient (first-order derivative), but also the curvature (second-order derivative) of the target distribution. Experimental results on synthetic models highlight the benefits of our approach in terms of step-size selection and posterior approximation accuracy when compared to other proposals.

Conor Rosato, Vincent Beraud, Paul Horridge et al.

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.