Scientists often run experiments to distinguish competing theories. This requires patience, rigor, and ingenuity - there is often a large space of possible experiments one could run. But we need not comb this space by hand - if we represent our theories as formal models and explicitly declare the space of experiments, we can automate the search for good experiments, looking for those with high expected information gain. Here, we present a general and principled approach to experiment design based on probabilistic programming languages (PPLs). PPLs offer a clean separation between declaring problems and solving them, which means that the scientist can automate experiment design by simply declaring her model and experiment spaces in the PPL without having to worry about the details of calculating information gain. We demonstrate our system in two case studies drawn from cognitive psychology, where we use it to design optimal experiments in the domains of sequence prediction and categorization. We find strong empirical validation that our automatically designed experiments were indeed optimal. We conclude by discussing a number of interesting questions for future research.
The basic intuition behind OED---to find experiments that maximize some expected measure of informativeness---has been independently discovered in a number of fields, including physics @cite_7 , chemistry @cite_2 , biology @cite_12 @cite_4 , psychology @cite_9 , statistics @cite_0 , and machine learning @cite_8 .
@cite_4 @cite_7 @cite_8 @cite_9 @cite_0 @cite_2 @cite_12
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SUMMARY When designing an experiment, the aim is usually to find the design which minimizes expected post-experimental uncertainties on the model parameters. Classical methods for experimental design are shown to fail in nonlinear problems because they incorporate linearized design criteria. A more fundamental criterion is introduced which, in principle, can be used to design any nonlinear problem. The criterion is entropy-based and depends on the calculation of marginal probability distributions. In turn, this requires the numerical calculation of integrals for which we use Monte Carlo sampling. The choice of discretization in the parameter data space strongly influences the number of samples required. Thus, the only practical limitation for this technique appears to be computational power. A synthetic experiment with an oscillatory, highly nonlinear parameter‐data relationship and a simple seismic amplitude versus offset (AVO) experiment are used to demonstrate the method. Interestingly, in our AVO example, although overly coarse discretizations lead to incorrect evaluation of the entropy, the optimal design remains unchanged. We tackle the fundamental problem of Bayesian active learning with noise, where we need to adaptively select from a number of expensive tests in order to identify an unknown hypothesis sampled from a known prior distribution. In the case of noise-free observations, a greedy algorithm called generalized binary search (GBS) is known to perform near-optimally. We show that if the observations are noisy, perhaps surprisingly, GBS can perform very poorly. We develop EC2, a novel, greedy active learning algorithm and prove that it is competitive with the optimal policy, thus obtaining the first competitiveness guarantees for Bayesian active learning with noisy observations. Our bounds rely on a recently discovered diminishing returns property called adaptive submodularity, generalizing the classical notion of submodular set functions to adaptive policies. Our results hold even if the tests have non-uniform cost and their noise is correlated. We also propose EffECXtive, a particularly fast approximation of EC2, and evaluate it on a Bayesian experimental design problem involving human subjects, intended to tease apart competing economic theories of how people make decisions under uncertainty. Models of a psychological process can be difficult to discriminate experimentally because it is not easy to determine the values of the critical design variables (e.g., presentation schedule, stimulus structure) that will be most informative in differentiating them. Recent developments in sampling-based search methods in statistics make it possible to determine these values, and thereby identify an optimal experimental design. After describing the method, it is demonstrated in two content areas in cognitive psychology in which models are highly competitive: retention (i.e., forgetting) and categorization. The optimal design is compared with the quality of designs used in the literature. The findings demonstrate that design optimization has the potential to increase the informativeness of the experimental method. The optimal selection of experimental conditions is essential in maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. A general Bayesian framework for optimal experimental design with nonlinear simulation-based models is proposed. The formulation accounts for uncertainty in model parameters, observables, and experimental conditions. Straightforward Monte Carlo evaluation of the objective function - which reflects expected information gain (Kullback-Leibler divergence) from prior to posterior - is intractable when the likelihood is computationally intensive. Instead, polynomial chaos expansions are introduced to capture the dependence of observables on model parameters and on design conditions. Under suitable regularity conditions, these expansions converge exponentially fast. Since both the parameter space and the design space can be high-dimensional, dimension-adaptive sparse quadrature is used to construct the polynomial expansions. Stochastic optimization methods will be used in the future to maximize the expected utility. While this approach is broadly applicable, it is demonstrated on a chemical kinetic system with strong nonlinearities. In particular, the Arrhenius rate parameters in a combustion reaction mechanism are estimated from observations of autoignition. Results show multiple order-of-magnitude speedups in both experimental design and parameter inference. Motivation: Systems biology employs mathematical modelling to further our understanding of biochemical pathways. Since the amount of experimental data on which the models are parameterized is often limited, these models exhibit large uncertainty in both parameters and predictions. Statistical methods can be used to select experiments that will reduce such uncertainty in an optimal manner. However, existing methods for optimal experiment design (OED) rely on assumptions that are inappropriate when data are scarce considering model complexity. Results: We have developed a novel method to perform OED for models that cope with large parameter uncertainty. We employ a Bayesian approach involving importance sampling of the posterior predictive distribution to predict the efficacy of a new measurement at reducing the uncertainty of a selected prediction. We demonstrate the method by applying it to a case where we show that specific combinations of experiments result in more precise predictions. Availability and implementation: Source code is available at: http: bmi.bmt.tue.nl sysbio software pua.html Contact:j.vanlier@tue.nl; N.A.W.v.Riel@tue.nl Supplementary information:Supplementary data are available at Bioinformatics online.