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Register a new userConsiderate Approaches to Achieving Sufficiency for ABC model selection
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Sarah Filippi
Publication date
2011
fields
Mathematical Statistics
and research's language is
English
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For nearly any challenging scientific problem evaluation of the likelihood is problematic if not impossible. Approximate Bayesian computation (ABC) allows us to employ the whole Bayesian formalism to problems where we can use simulations from a model, but cannot evaluate the likelihood directly. When summary statistics of real and simulated data are compared --- rather than the data directly --- information is lost, unless the summary statistics are sufficient. Here we employ an information-theoretical framework that can be used to construct (approximately) sufficient statistics by combining different statistics until the loss of information is minimized. Such sufficient sets of statistics are constructed for both parameter estimation and model selection problems. We apply our approach to a range of illustrative and real-world model selection problems.
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In this tutorial we schematically illustrate four algorithms: (1) ABC rejection for parameter estimation (2) ABC SMC for parameter estimation (3) ABC rejection for model selection on the joint space (4) ABC SMC for model selection on the joint space.
This paper is due to appear as a chapter of the forthcoming Handbook of Approximate Bayesian Computation (ABC) by S. Sisson, L. Fan, and M. Beaumont. We describe the challenge of calibrating climate simulators, and discuss the differences in emphasis in climate science compared to many of the more traditional ABC application areas. The primary difficulty is how to do inference with a computationally expensive simulator which we can only afford to run a small number of times, and we describe how Gaussian process emulators are used as surrogate models in this case. We introduce the idea of history matching, which is a non-probabilistic calibration method, which divides the parameter space into (not im)plausible and implausible regions. History matching can be shown to be a special case of ABC, but with a greater emphasis on defining realistic simulator discrepancy bounds, and using these to define tolerances and metrics. We describe a design approach for choosing parameter values at which to run the simulator, and illustrate the approach on a toy climate model, showing that with careful design we can find the plausible region with a very small number of model evaluations. Finally, we describe how calibrated GENIE-1 (an earth system model of intermediate complexity) predictions have been used, and why it is important to accurately characterise parametric uncertainty.
We compare two major approaches to variable selection in clustering: model selection and regularization. Based on previous results, we select the method of Maugis et al. (2009b), which modified the method of Raftery and Dean (2006), as a current state of the art model selection method. We select the method of Witten and Tibshirani (2010) as a current state of the art regularization method. We compared the methods by simulation in terms of their accuracy in both classification and variable selection. In the first simulation experiment all the variables were conditionally independent given cluster membership. We found that variable selection (of either kind) yielded substantial gains in classification accuracy when the clusters were well separated, but few gains when the clusters were close together. We found that the two variable selection methods had comparable classification accuracy, but that the model selection approach had substantially better accuracy in selecting variables. In our second simulation experiment, there were correlations among the variables given the cluster memberships. We found that the model selection approach was substantially more accurate in terms of both classification and variable selection than the regularization approach, and that both gave more accurate classifications than $K$-means without variable selection.
Approximate Bayesian computation (ABC) or likelihood-free inference algorithms are used to find approximations to posterior distributions without making explicit use of the likelihood function, depending instead on simulation of sample data sets from the model. In this paper we show that under the assumption of the existence of a uniform additive model error term, ABC algorithms give exact results when sufficient summaries are used. This interpretation allows the approximation made in many previous application papers to be understood, and should guide the choice of metric and tolerance in future work. ABC algorithms can be generalized by replacing the 0-1 cut-off with an acceptance probability that varies with the distance of the simulated data from the observed data. The acceptance density gives the distribution of the error term, enabling the uniform error usually used to be replaced by a general distribution. This generalization can also be applied to approximate Markov chain Monte Carlo algorithms. In light of this work, ABC algorithms can be seen as calibration techniques for implicit stochastic models, inferring parameter values in light of the computer model, data, prior beliefs about the parameter values, and any measurement or model errors.
This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described. Some of the most relevant methodologies are compared through theoretical comparisons and numerical experiments.
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