Florian Hartig

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Statistical inference for stochastic simulation models – theory and application

Florian Hartig, Justin M. Calabrese, Björn Reineking, Thorsten Wiegand & Andreas Huth

Ecology Letters, 14(8), 816–827 (2011)
Cite this
@article{hartig2011statistical,
  author = {Florian Hartig and Justin M. Calabrese and Björn Reineking and Thorsten Wiegand and Andreas Huth},
  title = {Statistical inference for stochastic simulation models – theory and application},
  journal = {Ecology Letters},
  volume = {14},
  number = {8},
  pages = {816--827},
  year = {2011},
  doi = {10.1111/j.1461-0248.2011.01640.x},
}

DOI: 10.1111/j.1461-0248.2011.01640.x
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Funding

This study was supported through funding by ERC advanced grant (FH) (233066).

View article (DOI)

Abstract

Statistical models are the traditional choice to test scientific theories when observations, processes or boundary conditions are subject to stochasticity. Many important systems in ecology and biology, however, are difficult to capture with statistical models. Stochastic simulation models offer an alternative, but they were hitherto associated with a major disadvantage: their likelihood functions can usually not be calculated explicitly, and thus it is difficult to couple them to well-established statistical theory such as maximum likelihood and Bayesian statistics. A number of new methods, among them Approximate Bayesian Computing and Pattern-Oriented Modelling, bypass this limitation. These methods share three main principles: aggregation of simulated and observed data via summary statistics, likelihood approximation based on the summary statistics, and efficient sampling. We discuss principles as well as advantages and caveats of these methods, and demonstrate their potential for integrating stochastic simulation models into a unified framework for statistical modelling.

What the paper shows and why it matters

(AI-generated)

Statistical inference in ecology and biology mostly requires a likelihood that can be calculated: the probability of the observed data for given parameter values. Systems in which many individuals interact by chance and are only partly observed usually lack such a likelihood and are modelled with stochastic simulation models. The likelihood of such a model usually cannot be calculated explicitly, so these models were seen as hard to connect to maximum likelihood and Bayesian statistics. Methods that get around this appear to have grown up in separate fields with little exchange. The paper takes on the assumption that a likelihood that cannot be calculated places simulation models outside likelihood-based statistical inference.

The paper argues that a stochastic simulation model does have a likelihood, which is only implicit (termed implicit statistical models by Diggle and Gratton 1984), and that the methods for connecting such models to data mostly follow the same three steps (not all approximate the likelihood explicitly). Estimating the likelihood by simulation alone is hopelessly inefficient in most practical cases, so the methods (1) reduce observed and simulated data to summary statistics, (2) approximate the likelihood of the observed statistics at a given parameter value, and (3) explore the parameter space efficiently, for example by Markov chain Monte Carlo. For step 2 the paper distinguishes four groups, from nonparametric methods such as Approximate Bayesian Computation, which accept simulations close to the observations, to rejection filters, as in Pattern-Oriented Modelling, which only sort parameter sets by tests against the data. Caveats include that the error of the approximation affects the sampler it is combined with, and that summary statistics adequate for estimating parameters may not be adequate for comparing models. A simulated movement path (summary statistics) and a brown bear model for the Eastern Alps, fitted by rejection of parameter sets (Pattern-Oriented Modelling), serve as examples.

The paper concludes that, once a likelihood approximation exists, fitting, comparing and checking the uncertainty of a simulation model can follow the logic of any statistical model, so simulation modelling can, in principle, join one statistical framework. It names usability and standardisation as the main open issues and hopes to prompt exchange across communities. Grimm and Railsback (2012) point to it for optimisation techniques when many parameters are calibrated.

References

  • Diggle & Gratton (1984). Monte Carlo Methods of Inference for Implicit Statistical Models. Journal of the Royal Statistical Society Series B: Statistical Methodology 46, 193–212. doi:10.1111/j.2517-6161.1984.tb01290.x
  • Grimm & Railsback (2012). Pattern-oriented modelling: a ‘multi-scope’ for predictive systems ecology. Philosophical Transactions of the Royal Society B: Biological Sciences 367, 298–310. doi:10.1098/rstb.2011.0180