Bayesian experimental design
Bayesian experimental design provides a general probability-theoretical 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 (e.g. 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.
Relations to more specialized optimal design 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.
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.
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.
Given a vector of parameters to determine, a prior PDF over those parameters and a PDF for making observation , given parameter values and an experiment design , the posterior PDF can be calculated using Bayes' theorem
where is the marginal probability density in observation space
The expected utility of an experiment with design can then be defined
where is some real-valued functional of the posterior PDF after making observation using an experiment design .
Gain in Shannon information as utility
Utility may be defined as the prior-posterior gain in Shannon information
Another possibility is to define the utility as
of which the latter can be evaluated without the need for evaluating individual posterior PDFs for all possible observations . It is worth noting that the first term on the second equation line will not depend on the design , as long as the observational uncertainty doesn't. On the other hand, the integral of in the first form is constant for all , 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, e.g. van den Berg, Curtis & Trampert (2003) and Ryan (2003). Note that
the expected information gain being exactly the mutual information between the parameter θ and the observation y. An example of Bayesian design for linear dynamical model discrimination is given in Bania (2019). Since was difficult to calculate, its lower bound has been used as a utility function. The lower bound is then maximized under the signal energy constraint. Proposed Bayesian design has been also compared with classical average D-optimal design. It was shown that the Bayesian design is superior to D-optimal design.
The Kelly criterion also describes such a utility function for a gambler seeking to maximize profit, which is used in gambling and information theory; Kelly's situation is identical to the foregoing, with the side information, or "private wire" taking the place of the experiment.
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- Vanlier; Tiemann; Hilbers; van Riel (2012), "A Bayesian approach to targeted experiment design" (PDF), Bioinformatics, 28 (8): 1136–1142, doi:10.1093/bioinformatics/bts092, PMC 3324513, PMID 22368245
- Liepe; Filippi; Komorowski; Stumpf (2013), "Maximizing the Information Content of Experiments in Systems Biology", PLOS Computational Biology, 9 (1): e1002888, doi:10.1371/journal.pcbi.1002888, PMC 3561087, PMID 23382663
- 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:10.1046/j.1365-246x.2003.02048.x, archived from the original (PDF) on 2011-07-17
- Chaloner, Kathryn; Verdinelli, Isabella (1995), "Bayesian experimental design: a review" (PDF), Statistical Science, 10 (3): 273–304, doi:10.1214/ss/1177009939
- DasGupta, A. (1996), "Review of optimal Bayes designs" (PDF), in Ghosh, S.; Rao, C. R. (eds.), Design and Analysis of Experiments, Handbook of Statistics, 13, North-Holland, pp. 1099–1148, ISBN 978-0-444-82061-7
- Lindley, D. V. (1956), "On a measure of information provided by an experiment", Annals of Mathematical Statistics, 27 (4): 986–1005, doi:10.1214/aoms/1177728069
- Ryan, K. J. (2003), "Estimating Expected Information Gains for Experimental Designs With Application to the Random Fatigue-Limit Model", Journal of Computational and Graphical Statistics, 12 (3): 585–603, doi:10.1198/1061860032012
- Bania, P. (2019), "Bayesian Input Design for Linear Dynamical Model Discrimination", Entropy, 21 (4), doi:10.3390/e21040351