# Efficient designs for Bayesian networks with sub-tree bounds

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## Abstract

We present upper and lower bounds for information measures, and use these to find the optimal design of experiments for Bayesian networks. The bounds are inspired by properties of the junction tree algorithm, which is commonly used for calculating conditional probabilities in graphical models like Bayesian networks. We demonstrate methods for iteratively improving the upper and lower bounds until they are sufficiently tight. We illustrate properties of the algorithm by tutorial examples in the case where we want to ensure optimality and for the case where the goal is an approximate solution with a guarantee. We further use the bounds to accelerate established algorithms for constructing useful designs. An example with petroleum fields in the North Sea is studied, where the design problem is related to exploration drilling campaigns. All of our examples consider binary random variables, but the theory can also be applied to other discrete or continuous distributions.

## Keywords

Information measure Bayesian networks Upper and lower bounds Almond tree Junction tree algorithm Petroleum exploration Subset selection Design of experiment## Notes

### Acknowledgments

This work is funded by Statistics for Innovation, \((\text {sfi})^2\), one of the Norwegian Centres for Research-based Innovation.

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