Bayesian experimental design
bayesian experimental design examples, bayesian experimental design diagramBayesian experimental design provides a general probabilitytheoretical framework from which other theories on experimental design can be derived It is based on Bayesian inference to interpret the observations/data acquired during the experiment This allows accounting for both any prior knowledge on the parameters to be determined as well as uncertainties in observations
The theory of Bayesian experimental design is to a certain extent based on the theory for making optimal decisions under uncertainty The aim when designing an experiment is to maximize the expected utility of the experiment outcome The utility is most commonly defined in terms of a measure of the accuracy of the information provided by the experiment eg the Shannon information or the negative variance, but may also involve factors such as the financial cost of performing the experiment What will be the optimal experiment design depends on the particular utility criterion chosen
Contents
 1 Relations to more specialized optimal design theory
 11 Linear theory
 12 Approximate normality
 13 Posterior distribution
 2 Mathematical formulation
 21 Gain in Shannon information as utility
 3 See also
 4 References
Relations to more specialized optimal design theory
Linear theory
If the model is linear, the prior probability density function PDF is homogeneous and observational errors are normally distributed, the theory simplifies to the classical optimal experimental design theory
Approximate normality
In numerous publications on Bayesian experimental design, it is often implicitly assumed that all posterior PDFs will be approximately normal This allows for the expected utility to be calculated using linear theory, averaging over the space of model parameters, an approach reviewed in Chaloner & Verdinelli 1995 Caution must however be taken when applying this method, since approximate normality of all possible posteriors is difficult to verify, even in cases of normal observational errors and uniform prior PDF
Posterior distribution
Recently, increased computational resources allow inference of the posterior distribution of model parameters, which can directly be used for experiment design Vanlier et al 2012 proposed an approach that uses the posterior predictive distribution to assess the effect of new measurements on prediction uncertainty, while Liepe et al 2013 suggest maximizing the mutual information between parameters, predictions and potential new experiments
Mathematical formulation
Notation

Given a vector θ of parameters to determine, a prior PDF p θ over those parameters and a PDF p y  θ , ξ for making observation y , given parameter values θ and an experiment design ξ , the posterior PDF can be calculated using Bayes' theorem
p θ  y , ξ = p y  θ , ξ p θ p y  ξ , }\,,}where p y  ξ is the marginal probability density in observation space
p y  ξ = ∫ p θ p y  θ , ξ d θ \,}The expected utility of an experiment with design ξ can then be defined
U ξ = ∫ p y  ξ U y , ξ d y , \,,}where U y , ξ is some realvalued functional of the posterior PDF p θ  y , ξ after making observation y using an experiment design ξ
Gain in Shannon information as utility
Utility may be defined as the priorposterior gain in Shannon information
U y , ξ = ∫ log p θ  y , ξ p θ  y , ξ d θ − ∫ log p θ p θ d θ \int \,}Another possibility is to define the utility as
U y , ξ = D K L p θ  y , ξ ‖ p θ , p\theta y,\xi \p\theta \,,}the Kullback–Leibler divergence of the prior from the posterior distribution Lindley 1956 noted that the expected utility will then be coordinateindependent and can be written in two forms
U ξ = ∫ ∫ log p θ  y , ξ p θ , y  ξ d θ d y − ∫ log p θ p θ d θ = ∫ ∫ log p y  θ , ξ p θ , y  ξ d y d θ − ∫ log p y  ξ p y  ξ d y , U\xi &=\int dy}\int \\&=\int d\theta }\int ,\end}\,}of which the latter can be evaluated without the need for evaluating individual posterior PDFs p θ  y , ξ for all possible observations y It is worth noting that the first term on the second equation line will not depend on the design ξ , as long as the observational uncertainty doesn't On the other hand, the integral of p θ log p θ in the first form is constant for all ξ , so if the goal is to choose the design with the highest utility, the term need not be computed at all Several authors have considered numerical techniques for evaluating and optimizing this criterion, eg van den Berg, Curtis & Trampert 2003 and Ryan 2003 Note that
U ξ = I θ ; y ,the expected information gain being exactly the mutual information between the parameter θ and the observation y Kelly 1956 also derived just such a utility function for a gambler seeking to profit maximally from side information in a horse race; Kelly's situation is identical to the foregoing, with the side information, or "private wire" taking the place of the experiment
See also
 Bayesian optimization
 Optimal Designs
 Active Learning
This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations Please help to improve this article by introducing more precise citations March 2011 Learn how and when to remove this template message 
References
 Vanlier; Tiemann; Hilbers; van Riel 2012, "A Bayesian approach to targeted experiment design" PDF, Bioinformatics, 28 8: 1136–1142, doi:101093/bioinformatics/bts092
 Liepe; Filippi; Komorowski; Stumpf 2013, "Maximizing the Information Content of Experiments in Systems Biology", PLOS Computational Biology, 9 1: e1002888, doi:101371/journalpcbi1002888
 van den Berg; Curtis; Trampert 2003, "Optimal nonlinear Bayesian experimental design: an application to amplitude versus offset experiments" PDF, Geophysical Journal International, 155 2: 411–421, doi:101046/j1365246x200302048x, archived from the original PDF on 20110717
 Chaloner, Kathryn; Verdinelli, Isabella 1995, "Bayesian experimental design: a review" PDF, Statistical Science, 10 3: 273–304, doi:101214/ss/1177009939
 DasGupta, A 1996, "Review of optimal Bayes designs", in Ghosh, S; Rao, C R, Design and Analysis of Experiments PDF, Handbook of Statistics, 13, NorthHolland, pp 1099–1148, ISBN 0444820612
 Lindley, D V 1956, "On a measure of information provided by an experiment", Annals of Mathematical Statistics, 27 4: 986–1005, doi:101214/aoms/1177728069
 Ryan, K J 2003, "Estimating Expected Information Gains for Experimental Designs With Application to the Random FatigueLimit Model", Journal of Computational and Graphical Statistics, 12 3: 585–603, doi:101198/1061860032012



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