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 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 for all possible observations . It is worth noting that the second 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. 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 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. See also Bayesian optimization Optimal design Active Learning References Bayesian statistics Design of experiments Optimal decisions Industrial engineering Systems engineering