Bayesian experimental design provides a general probability-theoretical framework from which other theories onexperimental design can be derived. It is based onBayesian 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^[1] is to a certain extent based on the theory for makingoptimal 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., theShannon information or the negative of thevariance) 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.

If the model is linear, theprior probability density function (PDF) is homogeneous and observational errors arenormally distributed, the theory simplifies to the classicaloptimal experimental design theory.

In numerous publications on Bayesian experimental design, it is (often implicitly) assumed that allposterior probabilities will be approximately normal. This allows for the expected utility to be calculated using linear theory, averaging over the space of model parameters.^[2] 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 probability.

In many cases, the posterior distribution is not available in closed form and has to be approximated using numerical methods. The most common approach is to useMarkov chain Monte Carlo methods to generate samples from the posterior, which can then be used to approximate the expected utility.

Another approach is to use avariational Bayes approximation of the posterior, which can often be calculated in closed form. This approach has the advantage of being computationally more efficient than Monte Carlo methods, but the disadvantage that the approximation might not be very accurate.

Some authors proposed approaches that use theposterior predictive distribution to assess the effect of new measurements on prediction uncertainty,^[3][4] while others suggest maximizing the mutual information between parameters, predictions and potential new experiments.^[5]

Notation ${\displaystyle \theta }$ parameters to be determined ${\displaystyle y}$ observation or data ${\displaystyle \xi }$ design ${\displaystyle p(y\mid \theta ,\xi )}$ PDF for making observation${\displaystyle y}$, given parameter values${\displaystyle \theta }$ and design${\displaystyle \xi }$ ${\displaystyle p(\theta )}$ prior PDF ${\displaystyle p(y\mid \xi )}$ marginal PDF in observation space ${\displaystyle p(\theta \mid y,\xi )}$    posterior PDF ${\displaystyle U(\xi )}$    utility of the design${\displaystyle \xi }$ ${\displaystyle U(y,\xi )}$    utility of the experiment outcome after observation${\displaystyle y}$ with design${\displaystyle \xi }$

Given a vector${\displaystyle \theta }$ of parameters to determine, a prior probability${\displaystyle p(\theta )}$ over those parameters and alikelihood${\displaystyle p(y\mid \theta ,\xi )}$ for making observation${\displaystyle y}$, given parameter values${\displaystyle \theta }$ and an experiment design${\displaystyle \xi }$, the posterior probability can be calculated usingBayes' theorem

${\displaystyle p(\theta \mid y,\xi )=\frac{p(y\mid \theta ,\xi )p(\theta )}{p(y\mid \xi )},}$

where${\displaystyle p(y\mid \xi )}$ is the marginal probability density in observation space

${\displaystyle p(y\mid \xi )=\int p(\theta )p(y\mid \theta ,\xi )d\theta .}$

The expected utility of an experiment with design${\displaystyle \xi }$ can then be defined

${\displaystyle U(\xi )=\int p(y\mid \xi )U(y,\xi )dy,}$

where${\displaystyle U(y,\xi )}$ is some real-valued functional of the posterior probability${\displaystyle p(\theta \mid y,\xi )}$ after making observation${\displaystyle y}$ using an experiment design${\displaystyle \xi }$.

Utility may be defined as the prior-posterior gain inShannon information

${\displaystyle U(y,\xi )=\int \log (p(\theta \mid y,\xi ))p(\theta |y,\xi )d\theta -\int \log (p(\theta ))p(\theta )d\theta .}$

Another possibility is to define the utility as

${\displaystyle U(y,\xi )=D_{KL}(p(\theta \mid y,\xi )\Vert p(\theta )),}$

theKullback–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 probability${\displaystyle p(\theta \mid y,\xi )}$ for all possible observations${\displaystyle y}$.^[6] It is worth noting that the second term on the second equation line will not depend on the design${\displaystyle \xi }$, as long as the observational uncertainty doesn't. On the other hand, the integral of${\displaystyle p(\theta )\log p(\theta )}$ in the first form is constant for all${\displaystyle \xi }$, 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.^[7][8] Note that

${\displaystyle U(\xi )=I(\theta ;y),}$

the expectedinformation gain being exactly themutual information between the parameterθ and the observationy. An example of Bayesian design for linear dynamical model identification are given in^[9][10].

Since${\displaystyle I(\theta ;y),}$ 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.

TheKelly criterion also describes such a utility function for a gambler seeking to maximize profit, which is used ingambling and information theory; Kelly's situation is identical to the foregoing, with the side information, or "private wire" taking the place of the experiment.

• Bayesian optimization
• Optimal design
• Active Learning
• Expected value of sample information

• DasGupta, A. (1996),"Review of optimal Bayes designs"(PDF), in Ghosh, S.;Rao, C. R. (eds.),Design and Analysis of Experiments, Handbook of Statistics, vol. 13, North-Holland, pp. 1099–1148,ISBN 978-0-444-82061-7
• Rainforth, Tom; et al. (2023),Modern Bayesian Experimental Design,arXiv:2302.14545

• Bayesian statistics
• Design of experiments
• Optimal decisions
• Industrial engineering
• Systems engineering

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