﻿ Bayesian experimental design
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# Bayesian experimental design

bayesian experimental design examples, bayesian experimental design diagram
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 eg 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

## Contents

• 1 Relations to more specialized optimal design theory
• 11 Linear theory
• 12 Approximate normality
• 13 Posterior distribution
• 2 Mathematical formulation
• 21 Gain in Shannon information as utility
• 4 References

## 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 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

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

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

Given a vector θ of parameters to determine, a prior PDF p θ over those parameters and a PDF p y | θ , ξ for making observation y , given parameter values θ and an experiment design ξ , the posterior PDF can be calculated using Bayes' theorem

p θ | y , ξ = p y | θ , ξ p θ p y | ξ , }\,,}

where p y | ξ is the marginal probability density in observation space

p y | ξ = ∫ p θ p y | θ , ξ d θ \,}

The expected utility of an experiment with design ξ can then be defined

U ξ = ∫ p y | ξ U y , ξ d y , \,,}

where U y , ξ is some real-valued functional of the posterior PDF p θ | y , ξ after making observation y using an experiment design ξ

### Gain in Shannon information as utility

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

U y , ξ = ∫ log ⁡ p θ | y , ξ p θ | y , ξ d θ − ∫ log ⁡ p θ p θ d θ -\int \,}

Another possibility is to define the utility as

U y , ξ = D K L p θ | y , ξ ‖ p θ , p\theta |y,\xi \|p\theta \,,}

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

U ξ = ∫ ∫ log ⁡ p θ | y , ξ p θ , y | ξ d θ d y − ∫ log ⁡ p θ p θ d θ = ∫ ∫ log ⁡ p y | θ , ξ p θ , y | ξ d y d θ − ∫ log ⁡ p y | ξ p y | ξ d y , U\xi &=\int dy}-\int \\&=\int d\theta }-\int ,\end}\,}

of which the latter can be evaluated without the need for evaluating individual posterior PDFs p θ | y , ξ for all possible observations y It is worth noting that the first term on the second equation line will not depend on the design ξ , as long as the observational uncertainty doesn't On the other hand, the integral of p θ log ⁡ p θ in the first form is constant for all ξ , 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, eg van den Berg, Curtis & Trampert 2003 and Ryan 2003 Note that

U ξ = I θ ; y ,

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

• Bayesian optimization
• Optimal Designs
• Active Learning

## References

• Vanlier; Tiemann; Hilbers; van Riel 2012, "A Bayesian approach to targeted experiment design" PDF, Bioinformatics, 28 8: 1136–1142, doi:101093/bioinformatics/bts092
• Liepe; Filippi; Komorowski; Stumpf 2013, "Maximizing the Information Content of Experiments in Systems Biology", PLOS Computational Biology, 9 1: e1002888, doi:101371/journalpcbi1002888
• van den Berg; Curtis; Trampert 2003, "Optimal nonlinear Bayesian experimental design: an application to amplitude versus offset experiments" PDF, Geophysical Journal International, 155 2: 411–421, doi:101046/j1365-246x200302048x, archived from the original PDF on 2011-07-17
• Chaloner, Kathryn; Verdinelli, Isabella 1995, "Bayesian experimental design: a review" PDF, Statistical Science, 10 3: 273–304, doi:101214/ss/1177009939
• DasGupta, A 1996, "Review of optimal Bayes designs", in Ghosh, S; Rao, C R, Design and Analysis of Experiments PDF, Handbook of Statistics, 13, North-Holland, pp 1099–1148, ISBN 0-444-82061-2
• Lindley, D V 1956, "On a measure of information provided by an experiment", Annals of Mathematical Statistics, 27 4: 986–1005, doi:101214/aoms/1177728069
• 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:101198/1061860032012

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29.10.2014

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