The Lifebelt Particle Filter for robust estimation from low-valued count data

Particle filtering methods are well developed for continuous state-space models. When dealing with discrete spaces on bounded domains, particle filtering methods can still be applied to sample from and marginalise over the unknown hidden states. Nevertheless, problems such as particle degradation can arise in this context and be even more severe than they are within the continuous-state domain: proposed particles can easily be incompatible with the data and the discrete system could often result in all particles having weights of zero. However, if the boundaries of the discrete hidden space are known, then these could be used to prevent particle collapse. In this paper we introduce the Lifebelt Particle Filter (LBPF), a novel method for robust likelihood estimation when low-valued count data arise. The LBPF combines a standard particle filter with one (or more) \textit{lifebelt particles} which, by construction, will tend not to be incompatible with the data. A mixture of resampled and non-resampled particles allows for the preservation of the lifebelt particle, which, together with the remaining particle swarm, provides samples from the filtering distribution, and can be used to generate estimates of the likelihood. The LBPF can be used within a pseudo-marginal scheme to draw inference on static parameters, $ \boldsymbol{\theta} $, governing a discrete state-space model with low-valued counts. We present here the applied case estimating a parameter governing probabilities and timings of deaths and recoveries of hospitalised patients during an epidemic.