23,262 research outputs found
A variational Bayesian method for inverse problems with impulsive noise
We propose a novel numerical method for solving inverse problems subject to
impulsive noises which possibly contain a large number of outliers. The
approach is of Bayesian type, and it exploits a heavy-tailed t distribution for
data noise to achieve robustness with respect to outliers. A hierarchical model
with all hyper-parameters automatically determined from the given data is
described. An algorithm of variational type by minimizing the Kullback-Leibler
divergence between the true posteriori distribution and a separable
approximation is developed. The numerical method is illustrated on several one-
and two-dimensional linear and nonlinear inverse problems arising from heat
conduction, including estimating boundary temperature, heat flux and heat
transfer coefficient. The results show its robustness to outliers and the fast
and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy
Qualitative Robustness in Bayesian Inference
The practical implementation of Bayesian inference requires numerical
approximation when closed-form expressions are not available. What types of
accuracy (convergence) of the numerical approximations guarantee robustness and
what types do not? In particular, is the recursive application of Bayes' rule
robust when subsequent data or posteriors are approximated? When the prior is
the push forward of a distribution by the map induced by the solution of a PDE,
in which norm should that solution be approximated? Motivated by such
questions, we investigate the sensitivity of the distribution of posterior
distributions (i.e. posterior distribution-valued random variables, randomized
through the data) with respect to perturbations of the prior and data
generating distributions in the limit when the number of data points grows
towards infinity
Approximate Models and Robust Decisions
Decisions based partly or solely on predictions from probabilistic models may
be sensitive to model misspecification. Statisticians are taught from an early
stage that "all models are wrong", but little formal guidance exists on how to
assess the impact of model approximation on decision making, or how to proceed
when optimal actions appear sensitive to model fidelity. This article presents
an overview of recent developments across different disciplines to address
this. We review diagnostic techniques, including graphical approaches and
summary statistics, to help highlight decisions made through minimised expected
loss that are sensitive to model misspecification. We then consider formal
methods for decision making under model misspecification by quantifying
stability of optimal actions to perturbations to the model within a
neighbourhood of model space. This neighbourhood is defined in either one of
two ways. Firstly, in a strong sense via an information (Kullback-Leibler)
divergence around the approximating model. Or using a nonparametric model
extension, again centred at the approximating model, in order to `average out'
over possible misspecifications. This is presented in the context of recent
work in the robust control, macroeconomics and financial mathematics
literature. We adopt a Bayesian approach throughout although the methods are
agnostic to this position
Reference priors for high energy physics
Bayesian inferences in high energy physics often use uniform prior
distributions for parameters about which little or no information is available
before data are collected. The resulting posterior distributions are therefore
sensitive to the choice of parametrization for the problem and may even be
improper if this choice is not carefully considered. Here we describe an
extensively tested methodology, known as reference analysis, which allows one
to construct parametrization-invariant priors that embody the notion of minimal
informativeness in a mathematically well-defined sense. We apply this
methodology to general cross section measurements and show that it yields
sensible results. A recent measurement of the single top quark cross section
illustrates the relevant techniques in a realistic situation
- …