- POMP inference: other approaches complementing likelihood-based analysis
Edward Ionides
4/4/2016
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Objectives
To introduce various inference techniques for nonlinear POMP models as alternatives to likelihood-based inference.
To undestand how these alternative techniques can be used to complement the use of likelihood based inference, despite their loss of statistical efficiency and/or additional assumptions.
To provide some introduction to how these analyses can be carried out using the pomp package.
Just as for our treatment of ARMA models, and other linear Gaussian time series models, we’re focusing on likelihood-based inference for nonlinear POMP models in this course. However, if you feel a need or desire to investigate alternatives, this chapter describes some options.
14.1 Introduction
Many, many statistical methods have been proposed for inference on POMP models. Relevant literature surveys are given in He et al. (2010) and King et al. (in press).
The volume of research indicates both the importance and the difficulty of the problem.
Let’s start by considering three criteria to categorize inference methods: the plug-and-play property; full-information or feature-based; frequentist or Bayesian.
14.2 Plug-and-play methods
Inference methodology that calls ‘rprocess’ but not ‘dprocess’ is said to be plug-and-play. All popular modern Monte Carlo methods for POMP models fall into this category.
Simulation-based is another name for to plug-and-play.
Historically, simulation-based meant simulating forward from initial conditions to the end of the time series.
However, particle filtering methods instead consider each observation interval sequentially. They carry out multiple, carefully selected, simulations over each interval.
We permit plug-and-play methods to call ‘dmeasure’. A method that uses only ‘rprocess’ and ‘rmeasure’ is called doubly plug-and-play.
Two non-plug-and-play methods (EM algorithms and MCMC) have theoretical convergence problems for nonlinear POMP models. The failures of these two workhorses of statistical computation have prompted development of alternative methodology.
14.2.1 Full-information and feature-based methods
Full-information methods are defined to be those based on the likelihood function for the full data (i.e., likelihood-based frequentist inference and Bayesian inference).
Feature-based methods either consider a summary statistic (a function of the data) or work with an an alternative to the likelihood.
Asymptotically, full-information methods are statistically efficient and feature-based methods are not.
Loss of statistical efficiency could potentially be an acceptable tradeoff for advantages in computational efficiency.
However, good low-dimensional summary statistics can be hard to find.
When using statistically inefficient methods, it can be hard to know how much information you are losing.
Intuition and scientific reasoning can be inadequate tools to derive informative low-dimensional summary statistics (Ionides 2011, Shrestha et al. 2011).
With full-information methods, it can be hard to work out which feature, or features, of the data have strong influence on the conclusions of the data analysis.
Fitting the model to some selected feature, or features, can help establish whether those aspects of the data are informative about parameters of interest, and whether these features are in agreement with the rest of the data.
14.3 Bayesian and frequentist methods
Recently, plug-and-play Bayesian methods have been discovered:
particle Markov chain Monte Carlo (PMCMC) (Andrieu et al. 2010).
approximate Bayesian computation (ABC) (Toni et al. 2009).
Prior belief specification is both the strength and weakness of Bayesian methodology:
The likelihood surface for nonlinear POMP models often contains nonlinear ridges and variations in curvature.
These situations bring into question the appropriateness of independent priors derived from expert opinion on marginal distributions of parameters.
They also are problematic for specification of “flat” or “uninformative” prior beliefs.
Expert opinion “prior beliefs” can be also be introduced into non-Bayesian analysis, by treating it as additional data. However, our primary task is to identify the information in the data under investigation, so it can be helpful to use methods that do not force us to make our conclusions dependent on quantification of prior beliefs.
A good general reference on likelihood and its role in scientific inference is Pawitan (2001).
14.4 Full-information plug-and-play frequentist methods
Iterated filtering methods (Ionides et al. 2006, 2015) are the only currently available, full-information, plug-and-play, frequentist methods for POMP models.
Iterated filtering methods have been shown to solve likelihood-based inference problems for epidemiological situations which are computationally intractable for available Bayesian methodology.
Some previous applications of iterated filtering are listed on its Wikipedia article.
14.5 Summary of POMP inference methodologies
| Frequentist | Bayesian | ||
|---|---|---|---|
| Plug-and-play | Full-information | iterated filtering | particle MCMC |
| Feature-based | simulated moments | ABC | |
| synthetic likelihood | |||
| Not-plug-and-play | Full-information | EM algorithm | MCMC |
| Kalman filter | |||
| Feature-based | Yule-Walker | extended Kalman filter | |
| extended Kalman filter |
Yule-Walker is the method of moments for ARMA models. A Gaussian ARMA model can be represented as a linear Gaussian POMP.
The Kalman filter gives the exact likelihood for a linear Gaussian POMP. The extended Kalman filter gives an approximation for nonlinear models that can be used for quasi-likelihood or quasi-Bayesian inference.
14.6 POMP inference methodologies in pomp
| Methodology | pomp function |
|---|---|
| iterated filtering | mif2() |
| particle Markov chain Monte Carlo | pmcmc() |
| approximate Bayesian computing | abc() |
| feature-based synthetic likelihood | probe.match() |
| nonlinear forecasting | nlf() |
We won’t have time to demonstrate these methods.
Methodology descriptions, and examples using pomp, can be found in King et al, in press.
Acknowledgment
These notes draw on material developed for a short course on Simulation-based Inference for Epidemiological Dynamics by Aaron King and Edward Ionides, taught at the University of Washington Summer Institute in Statistics and Modeling in Infectious Diseases, 2015.
References
Andrieu, C., A. Doucet, and R. Holenstein. 2010. Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B 72:269–342.
He, D., E. L. Ionides, and A. A. King. 2010. Plug-and-play inference for disease dynamics: Measles in large and small populations as a case study. Journal of The Royal Society Interface 7:271–283.
Ionides, E. L. 2011. Discussion on “Feature matching in time series modeling” by Y. Xia and H. Tong. Statistical Science 26:49–52.
Ionides, E. L., C. Bretó, and A. A. King. 2006. Inference for nonlinear dynamical systems. Proceedings of the National Academy of Sciences of the U.S.A. 103:18438–18443.
Ionides, E. L., D. Nguyen, Y. Atchadé, S. Stoev, and A. A. King. 2015. Inference for dynamic and latent variable models via iterated, perturbed Bayes maps. Proceedings of the National Academy of Sciences of the U.S.A. 112:719–724.
King, A. A., D. Nguyen, and E. L. Ionides. in press. Statistical inference for partially observed Markov processes via the R package pomp. Journal of Statistical Software.
Pawitan, Y. 2001. In all likelihood: Statistical modelling and inference using likelihood. Clarendon Press, Oxford.
Shrestha, S., A. A. King, and P. Rohani. 2011. Statistical inference for multi-pathogen systems. PLoS Computational Biology 7:e1002135.
Toni, T., D. Welch, N. Strelkowa, A. Ipsen, and M. P. Stumpf. 2009. Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. Journal of the Royal Society Interface 6:187–202.