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 of the 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.
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 . 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. proposed an approach that uses the posterior predictive distribution to assess the effect of new measurements on prediction uncertainty, while suggest maximizing the mutual information between parameters, predictions and potential new experiments.
Mathematical formulation.
Given a vector formula_1 of parameters to determine, a prior PDF formula_2 over those parameters and a PDF formula_3 for making observation formula_4, given parameter values formula_1 and an experiment design formula_6, the posterior PDF can be calculated using Bayes' theorem
where formula_8 is the marginal probability density in observation space
The expected utility of an experiment with design formula_6 can then be defined
where formula_12 is some real-valued functional of the posterior PDF formula_13 after making observation formula_4 using an experiment design formula_6.
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
the Kullback–Leibler divergence of the prior from the posterior distribution.
 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
formula_19 for all possible observations formula_4. It is worth noting that the first term on the second equation line will not depend on the design formula_6, as long as the observational uncertainty doesn't. On the other hand, the integral of formula_22 in the first form is constant for all formula_6, 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. and . 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 . Since formula_25 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. 