PAL versus SMC: Two Approaches in Compartmental Modeling

Abstract

Partially Observed Markov Process (POMP) models have been extensively employed in epidemiological modeling over the past several decades to understand disease patterns and inform policy-making. Although the observation density is typically assumed to be known or at least can be evaluated point-wisely, the latent Markovian models lack closed-form expressions for initial density and transition kernels, making it challenging to compute the likelihood for POMP models with complex latent models. Recently, a novel approach (Poisson Approximate Likelihood, PAL) was introduced by Whitehouse et al. (2023), which employs a Poisson approximation to posterior densities, offering a fast and consistent approximation for the likelihood function. Whitehouse et al. (2023) claimed that their method, along with its associated model, improved the maximum likelihood estimation compared to traditional sequential Monte Carlo (SMC) methods used in Stocks et al. (2018) when applied to the German rotavirus, by approximately 3200 log-likelihood units. However, our analysis of comparing two methods applied to the model for the same rotavirus dataset reveals that the improvement of 3200 log-likelihood units results from the use of two different datasets differed by a scaling factor. Moreover, although PAL and SMC are two ways of approximating the likelihood, within the framework of models compatible with PAL, when computation time is preferred, PAL is recommended but it may suffer from a likelihood shortfall, which can't be overcome in general. When the model is misspecified for certain time points, SMC may fail, and PAL is possible to approximate the likelihood. When computational time is not a critical factor and the model is correctly specified, sequential Monte Carlo methods are recommended.