Iterated filtering: New theory, algorithms and applications

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Partially observed Markov process models provide a general framework for formulating and answering questions about dynamic systems. Evaluation of the likelihood for these models can be formulated as a filtering problem. Iterated filtering algorithms carry out repeated sequential Monte Carlo filtering operations to maximize the likelihood. Current theory for iterated filtering justifies the parameter update at each iteration via a stochastic approximation to the first derivative of the log likelihood. Our new approach to iterated filtering theory and methodology draws on similarities with data cloning (i.e., methods where Markov chain Monte Carlo algorithms are applied to multiple copies of the data to provide likelihood-based inference). The relationship with iterated filtering is that each filtering iteration is analogous to creating a new clone of the data. This new theoretical perspective leads to novel novel algorithms. In the context of the previous stochastic approximation theory of iterated filtering, the new algorithms behave as though the intractable second derivative of the likelihood were known. Indeed, the proposed algorithm generates an estimate of the Fisher information as a bi-product. Preliminary results, on a simple ecological model and on a challenging inference problem arising from fitting a malaria transmission model to time series data, show that a new iterated filtering algorithm out-performs previous methods. As well as advancing methodological capabilities for time series analysis via mechanistic models, the investigators will develop applications to two other related classes of statistical problems: longitudinal data analysis via mechanistic models, and inference for complex dynamic data structures. As concrete examples, the investigators will study the use of iterated filtering techniques for (i) relating pathogen genetic sequence data to HIV transmission models; (ii) using longitudinal data to inform stochastic dynamic models of sexual behaviors related to HIV transmission; (iii) inference via summary statistics and pseudo likelihood criteria, with an application to partially observed dynamic network models. Many scientific challenges involve the study of nonlinear stochastic dynamic systems about which only noisy or incomplete measurements are available. Except when the system is small, state-of-the-art statistical methods are required to make efficient use of available data and to provide modeling flexibility that promotes model criticism. The novel iterated filtering algorithms developed by the investigators will be used to study disease transmission systems with the goal of informing policy for the detection, control and potential eradication of infectious diseases. The PI is already engaged in the interface between statistical methodology development, epidemiology and public policy. The proposed research will directly benefit understanding of malaria and HIV transmission, but will also provide methodological tools and case studies relevant to other disease systems. More broadly, the methodology developed will be applicable to inference problems for dynamic systems arising throughout the biological, physical, social, health and engineering sciences. Open source software for all the methodology developed will be included in the R package {pomp} (http://cran.r-project.org/web/packages/pomp) for which the PI is a co-developer. Advances in iterated filtering methodology will be disseminated as part of the PIs ongoing agenda to spread the use of formal statistic methods for partially observed dynamic systems.