Output-weighted optimal sampling for Bayesian regression and rare event statistics using few samples
This work addresses optimal experimental design for Bayesian regression and rare event statistics, offering a novel approach for high-dimensional problems, though it appears incremental as it builds on existing methods like model error minimization and mutual information maximization.
The paper tackles the problem of efficiently estimating the statistics of an unknown function with minimal evaluations by introducing an output-weighted sampling criterion that adaptively selects inputs from important regions of the parameter space, enabling application to high-dimensional inputs.
For many important problems the quantity of interest is an unknown function of the parameters, which is a random vector with known statistics. Since the dependence of the output on this random vector is unknown, the challenge is to identify its statistics, using the minimum number of function evaluations. This problem can been seen in the context of active learning or optimal experimental design. We employ Bayesian regression to represent the derived model uncertainty due to finite and small number of input-output pairs. In this context we evaluate existing methods for optimal sample selection, such as model error minimization and mutual information maximization. We show that for the case of known output variance, the commonly employed criteria in the literature do not take into account the output values of the existing input-output pairs, while for the case of unknown output variance this dependence can be very weak. We introduce a criterion that takes into account the values of the output for the existing samples and adaptively selects inputs from regions of the parameter space which have important contribution to the output. The new method allows for application to high-dimensional inputs, paving the way for optimal experimental design in high-dimensions.