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@PHDTHESIS{Bartuska:1011019,
author = {Bartuska, Arved},
othercontributors = {Tempone, Raul and Espath, Luis and Scheichl, Robert},
title = {{H}ierarchical methods for {B}ayesian optimal experimental
design},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {RWTH Aachen University},
reportid = {RWTH-2025-04489},
pages = {1 Online-Ressource : Illustrationen},
year = {2025},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University; Dissertation, RWTH Aachen University, 2025},
abstract = {Finding the optimal design of an experiment is a task
frequently encountered in science, engineering, medicine,
and many other fields. A well-designed experiment yields
highly informative data at minimal cost. Bayesian optimal
experimental design is a comprehensive method where a
utility function is optimized to learn unknown parameters
from observation data, and the optimization variables define
the experiment design. Our utility function of choice is the
expected information gain (EIG), quantifying the expected
reduction of uncertainty regarding the parameters to be
inferred by the experiment. This task typically necessitates
the numerical estimation of a nested integration problem of
the form $\int f\left(\int g(y,x)dx\right)dy$, where $f$ is
a nonlinear function and the evaluation of $g$ typically
involves the solution of a partial differential equation.
The focus of this thesis is on the design of statistical
estimators for nested integrals and their numerical
analysis. The dimension of both the inner and outer
integrand renders quadrature methods costly for many
real-world applications. We employ Monte Carlo (MC) methods,
where the estimation error is proportional to the estimator
variance and provide an overview of the MC method and
various variance reduction methods, such as importance
sampling, multilevel MC, and randomized quasi-MC (rQMC).
These estimators harness the hierarchical structure
introduced by the inner integral estimation and a finite
element approximation of the inner integrand. Next, we
introduce novel numerical estimators for nested integration
problems based on the rQMC method and a combination of rQMC
and multilevel methods. The nonlinear function $f$
separating the outer from the inner integrand in the EIG is
the logarithm, exhibiting a singularity at 0. Assuming
additive Gaussian noise in the experiment model, this
singularity renders traditional rQMC error bounds via the
Koksma--Hlawka inequality pointless, as they result in an
infinite upper bound. Then, we introduce a truncation scheme
of the Gaussian noise to establish rigorous error bounds,
affecting the estimator complexity only by multiplicative
logarithmic terms. These error bounds are employed to
optimize the required number of samples and discretization
accuracy in the finite element approximation for a given
error tolerance. Our estimators are subsequently applied to
estimate the EIG of an experiment, serving as the objective
function in the Bayesian optimal experimental design
problem. We also study the setting where there is nuisance
uncertainty in the experiment model. This scenario is
encountered when the experimenter is interested in reducing
only some sources of uncertainty in the model, and not
others. This latter type is called nuisance uncertainty and
the distinction between the two depends on the aims of the
experimenter. Evaluating the EIG in such cases typically
requires the numerical estimation of an additional nested
integration problem. We introduce novel numerical estimators
based on a small-noise approximation to marginalize the
nuisance parameters, combined with the Laplace approximation
and MC methods. The Laplace approximation is particularly
useful for the case of numerous experiments to address the
resulting concentration of measure. One estimator applies
the Laplace approximation directly to the inner integral,
resulting in an asymptotically biased, nonnested estimator.
Another estimator uses the Laplace approximation for
importance sampling, ensuring asymptotic consistency. These
estimators are applicable when the nuisance uncertainty is
small compared to the observation noise of the experiment.
Two more estimators based on Laplace approximations are then
introduced to address the case of large nuisance
uncertainty. The first of these estimators employs a nesting
of a Laplace approximation and the related Laplace's method
to result in a nonnested estimate, and the other involves
separate importance sampling schemes based on Laplace
approximations. The efficiency of these estimators is
demonstrated for EIG estimation of experiments involving
nuisance parameters. The design of the underlying
experiments is then optimized via grid search wherever
feasible or using a stochastic gradient descent method.},
cin = {118110 / 110000},
ddc = {510},
cid = {$I:(DE-82)118110_20190107$ / $I:(DE-82)110000_20140620$},
pnm = {GRK 2379 - GRK 2379: Hierarchische und hybride Ansätze
für moderne inverse Probleme (333849990)},
pid = {G:(GEPRIS)333849990},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2025-04489},
url = {https://publications.rwth-aachen.de/record/1011019},
}
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