Bayesian experimental design

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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[edit]

Linear theory[edit]

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[edit]

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[edit]

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[edit]

Notation
parameters to be determined
observation or data
design
PDF for making observation , given parameter values and design
prior PDF
marginal PDF in observation space
   posterior PDF
   utility of the design
   utility of the experiment outcome after observation with design

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[edit]

Utility may be defined as the prior-posterior gain in Shannon information

Note also that

is the Kullback–Leibler divergence of the prior from the posterior distribution. Lindley (1956) noted that the expected utility will then be coordinate-independent and can be written in two forms

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. 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[edit]

References[edit]

  • 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