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

Particle filtering methods can be applied to estimation problems in discrete spaces on bounded domains, to sample from and marginalise over unknown hidden states. As in continuous settings, problems such as particle degradation can arise: proposed particles can be incompatible with the data, lying in low probability regions or outside the boundary constraints, and the discrete system could result in all particles having weights of zero. In this paper we introduce the Lifebelt Particle Filter (LBPF), a novel method for robust likelihood estimation in low-valued count problems. The LBPF combines a standard particle filter with one (or more) lifebelt particles which, by construction, lie within the boundaries of the discrete random variables, and therefore are compatible 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 unbiased estimates of the likelihood. The main benefit of the LBPF is that only one or few, wisely chosen, particles are sufficient to prevent particle collapse. Differently from other methods, there is no need to increase the number of particles, and therefore the computational effort, in regions of the parameter space that generate less likely hidden states. The LBPF can be used within a pseudo-marginal scheme to draw inferences on static parameters, $ \boldsymbol{\theta} $, governing the system. We address here the estimation of a parameter governing probabilities of deaths and recoveries of hospitalised patients during an epidemic.