Iterated Filtering and Smoothing with Application to Infectious Disease Models.

Abstract

Partially observed Markov process (POMP) models are ubiquitous tools for modeling time series data in many fields including statistics, econometrics, ecology, and engineering. Because of incomplete measurements, and possibly weakly identifiable parameters, making inferences on POMP models can be challenging. Standard methods for inference (e.g., maximum likelihood) with restrictive assumptions of linear Gaussian models have often led to unsatisfactory results when the assumptions are violated. To relax these assumptions, this dissertation develops a class of simulation-based algorithms called iterated filtering and smoothing for POMP models. First, a novel filter, called Bayes map iterated filtering, is introduced. This filter recursively combines parameter perturbations with latent variable reconstruction, stochastically optimizing the approximated likelihood of latent variable models and providing an asymptotic guarantee of the performance of this inference methodology. Second, a fast, light-weight algorithm, called second-order iterated smoothing is proposed to improve on the convergence rate of the approach. The goal of this part is to demonstrate that by exploiting Fisher Information as a by-product of the inference methodology, one can theoretically achieve both statistical and computational efficiencies without sacrificing applicability to a general class of models. Third, a new technique for the proof of Bayes map iterated filtering algorithm, based on super-martingale inequality, is proposed. This approach with verifiable conditions is simpler than the previous approach and is generalizable to more sophisticated algorithms. Fourth, we validated the properties of the proposed methodologies through applying them to a challenging inference problem of fitting a malaria transmission model with control to time series data, finding substantial gains for our methods over current alternatives. Finally, a range of modern statistical methodologies for POMP modeling have been implemented in an open source R package, named pomp, to provide a flexible computational framework for the community.