Approximate Bayesian computation methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper we discuss and apply an approximate Bayesian computation (ABC) method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC gives information about the inferability of parameters and model sensitivity to changes in parameters, and tends to perform better than other ABC approaches. The algorithm is applied to several well known biological systems, for which parameters and their credible intervals are inferred. Moreover, we develop ABC SMC as a tool for model selection; given a range of different mathematical descriptions, ABC SMC is able to choose the best model using the standard Bayesian model selection apparatus.
Most dynamical system studied in the physical, life and social sciences and engineering are modelled by ordinary, delay or stochastic differential equations. However, for the vast majority of systems and particularly for biological systems, we lack reliable information about parameters and frequently have several competing models for the structure of the underlying equations. Moreover, the biological experimental data is often scarce and incomplete, and the likelihood surfaces of large models are complex. The analysis of such dynamical systems therefore requires new, more realistic quantitative and predictive models. Here we will develop novel statistical tools that allow us to analyze such data in terms of dynamical models by (i) providing estimates for model parameter values, and (ii) allowing us to compare the performance of different models in describing the overall data.
In the last decade extensive research has been conducted on estimating the parameters of deterministic systems. Much attention has been given to local and global nonlinear optimization methods (Mendes & Kell, 1998;Moles et al., 2003) and generally parameter estimation has been performed by maximum likelihood estimation (Timmer & Muller, 2004;Muller et al., 2004;Baker et al., 2005;Bortz & Nelson, 2006). The methods developed for ordinary differential equations have been extended to ordinary differential equations with time delays (Horbelt et al., 2002). Deterministic models have also been parameterized in a Bayesian framework using Bayesian hierarchical models (Putter et al., 2002;Banks et al., 2005;Huang et al., 2006). Simulated annealing, which attempts to avoid getting trapped in local minima, is another well known optimization algorithm that has been found successful in various applications (Kirkpatrick et al., 1983;Mendes & Kell, 1998). There are also several Monte Carlo based approaches applied to parameter estimation of deterministic (Battogtokh et al., 2002;Brown & Sethna, 2003) and stochastic (Sisson et al., 2007) systems. Parameter estimation for stochastic models has been extensively explored in financial mathematics (Johannes & Polson, 2005) and has been applied to biological systems in a frequentist maximum likelihood (Reinker et al., 2006) and Bayesian (Golightly & Wilkinson, 2005, 2006;Wilkinson, 2006) framework.
Most commonly, model selection has been performed by likelihood ratio tests (in case of nested models) or the Akaike information criterion (in case of non-nested models). Recently Bayesian methods have increasingly been coming into use. Vyshemirsky & Girolami (2008) have investigated different ways of computing Bayes factors for model selection of deterministic differential equation models, and Brown & Sethna (2003) have used the Bayesian information criterion. In population genetics, model selection has been performed using approximate Bayesian computation (ABC) in its basic rejection form (Zucknick, 2004;Wilkinson, 2007) and coupled with multinomial logistic regression (Beaumont, 2008a;Fagundes et al., 2007).
There is thus a wide variety of tools available for parameter estimation and, to a lesser extent, model selection. However, to our knowledge, no method available can be applied to all different kinds of modelling approaches (e.g. ordinary or stochastic differential equations with and without time delay) without substantial modification, estimate credible intervals, take incomplete or partially observed data as input, be employed for model selection, and reliably explore the whole parameter space without getting trapped in local extrema.
In this paper we apply an ABC method based on sequential Monte Carlo (SMC) to the parameter estimation and model selection problem for dynamical models. In ABC methods the evaluation of the likelihood is replaced by a simulation-based procedure (Pritchard et al., 1999;Beaumont et al., 2002;Marjoram et al., 2003;Sisson et al., 2007). We explore the information that ABC SMC gives about inferability of parameters and sensitivity of the model to parameter variation. Furthermore we compare the performance of ABC SMC to other approximate Bayesian computation methods. The method is illustrated on two simulated datasets (one from ecology and one from molecular systems biology), and real and simulated epidemiological datasets. As we will show, ABC SMC yields reliable parameter estimates with credible intervals; can be applied to different types of models (e.g. deterministic as well as stochastic models); is relatively computationally efficient (and easily parallelized); allows for discrimination among sets of candidate models in a formal Bayesian model-selection sense; and gives us an assessment of model and parameter sensitivity.
In this section we review and develop the theory underlying ABC with emphasis on applications to dynamical systems, before introducing a formal Bayesian model selection approach in an ABC context.
Approximate Bayesian Computation methods have been co
This content is AI-processed based on open access ArXiv data.