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