# On Disturbance State-Space Models and the Particle Marginal Metropolis--Hastings Sampler

We investigate nonlinear state-space models without a closed-form transition density, and propose re-expressing such models over their latent noise variables rather than their latent state variables. In doing so the tractable noise density emerges in place of the intractable transition density. For importance sampling methods such as the auxiliary particle filter (APF), this enables importance weights to be computed where they could not otherwise be. As case studies we take two multivariate marine biogeochemical models and perform state and parameter estimation using the particle marginal Metropolis-Hastings (PMMH) sampler, within which is an APF component. For that APF, we compare several proposal strategies over noise variables, all based on lookaheads with the unscented Kalman filter. These strategies are compared using Markov chain convergence and acceptance rates, along with a newly introduced conditional acceptance rate that is useful for assessing the impact of using an estimated, rather than exact, likelihood. Results indicate the utility of designing proposals over the noise variables, particularly for fast-mixing process models.

### Citation

L.M. Murray, E.M. Jones and J. Parslow (2013). On Disturbance State-Space Models and the Particle Marginal Metropolis--Hastings Sampler. *SIAM/ASA Journal of Uncertainty Quantification*. **1**(1):494--521. doi:10.1137/130915376.

```
@Article{Murray2013a,
title = {On Disturbance State-Space Models and the Particle Marginal {M}etropolis---{H}astings Sampler},
author = {Lawrence M. Murray and Emlyn M. Jones and John Parslow},
journal = {SIAM/ASA Journal of Uncertainty Quantification},
year = {2013},
volume = {1},
number = {1},
pages = {494--521},
doi = {10.1137/130915376},}
```