Bayesian experimental design provides a general probabilitytheoretical framework from which other theories on experimental design can me 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.
In Bayesian experimental design, 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
Notation

Linear theory
If the model is linear, the prior probability density function (PDF) is homogeneous and observational errors are Multivariate normal distribution  normally distributed, the theory simplifies to the classical optimal design  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 approximately normality of all possible posteriors is difficult to verify, even in cases of normal observational errors and uniform a prior PDF.
Mathematical formulation
Given a vector of parameters to determine, a Prior probability  prior PDF over those parameters and a PDF for making observation , given parameter values and design , the posterior PDF can be calculated using Bayes’ theorem
, 
(1) 
where is the marginal probability density in observation space
. 
(2) 
The expected utility of an experiment with design can then be defined
, 
(3) 
where is some realvalued functional of the Posterior probability  posterior PDF after making observation using an experiment design .
Gain in Shannon information as utility
If the utility is defined as the priorposterior gain in Differential entropy  Shannon information
. 
(4) 
Lindley (1956) noted that the expected utility will then be coordinateindependent 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 . Worth noting is that the first term on the second equation line will be a constant, as long as the observational uncertainty does not depend on the design . Several authors have considered numerical techniques for evaluating and optimizing this criterion, e.g. van den Berg, Curtis & Trampert (2003) and Ryan (2003).
© 2008 Emanuel Winterfors code can be used in comments: $latex p(\theta)$ gives 
Very interested in this topic for pharmaceutical development. Can you email me any examples where the Bayesian approach has been used in nonlinear model DOE. Thanks!!
Comment by Dave LeBlond — January 6, 2010 @ 5:24 am