Generalized Bayesian inference under prior-data conflict

This thesis is concerned with the generalisation of Bayesian
inference towards the use of imprecise or interval probability,
with a focus on model behaviour in case of prior-data conflict.
Bayesian inference is one of the main approaches to statistical
inference. It requires to express (subjective) knowledge on the
parameter(s) of interest not incorporated in the data by a
so-called prior distribution. All inferences are then based on the
so-called posterior distribution, the subsumption of prior
knowledge and the information in the data calculated via Bayes'
Rule. The adequate choice of priors has always been an intensive
matter of debate in the Bayesian literature. While a considerable
part of the literature is concerned with so-called non-informative
priors aiming to eliminate (or, at least, to standardise) the
influence of priors on posterior inferences, inclusion of specific
prior information into the model may be necessary if data are
scarce, or do not contain much information about the parameter(s)
of interest; also, shrinkage estimators, common in frequentist
approaches, can be considered as Bayesian estimators based on
informative priors. When substantial information is used to elicit
the prior distribution through, e.g, an expert's assessment, and
the sample size is not large enough to eliminate the influence of
the prior, prior-data conflict can occur, i.e., information from
outlier-free data suggests parameter values which are surprising
from the viewpoint of prior information, and it may not be clear
whether the prior specifications or the integrity of the data
collecting method (the measurement procedure could, e.g., be
systematically biased) should be questioned. In any case, such a
conflict should be reflected in the posterior, leading to very
cautious inferences, and most statisticians would thus expect to
observe, e.g., wider credibility intervals for parameters in case
of prior-data conflict. However, at least when modelling is based
on conjugate priors, prior-data conflict is in most cases
completely averaged out, giving a false certainty in posterior
inferences. Here, imprecise or interval probability methods offer
sound strategies to counter this issue, by mapping parameter
uncertainty over sets of priors resp. posteriors instead of over
single distributions. This approach is supported by recent research
in economics, risk analysis and artificial intelligence,
corroborating the multi-dimensional nature of uncertainty and
concluding that standard probability theory as founded on
Kolmogorov's or de Finetti's framework may be too restrictive,
being appropriate only for describing one dimension, namely ideal
stochastic phenomena. The thesis studies how to efficiently
describe sets of priors in the setting of samples from an
exponential family. Models are developed that offer enough
flexibility to express a wide range of (partial) prior information,
give reasonably cautious inferences in case of prior-data conflict
while resulting in more precise inferences when prior and data
agree well, and still remain easily tractable in order to be useful
for statistical practice. Applications in various areas, e.g.
common-cause failure modeling and Bayesian linear regression, are
explored, and the developed approach is compared to other imprecise
probability models.