Related papers: Well-posed Bayesian Inverse Problems with Infinite…
This paper presents an improved implicit sampling method for hierarchical Bayesian inverse problems. A widely used approach for sampling posterior distribution is based on Markov chain Monte Carlo (MCMC). However, the samples generated by…
In a Bayesian inverse problem setting, the solution consists of a posterior measure obtained by combining prior belief, information about the forward operator, and noisy observational data. This measure is most often given in terms of a…
We study the empirical version of halfspace depths with the objective of establishing a connection between the rates of convergence and the tail behaviour of the corresponding underlying distributions. The intricate interplay between the…
Bayesian Last Layer (BLL) models focus solely on uncertainty in the output layer of neural networks, demonstrating comparable performance to more complex Bayesian models. However, the use of Gaussian priors for last layer weights in…
Inverse problems involving partial differential equations (PDEs) are widely used in science and engineering. Although such problems are generally ill-posed, different regularisation approaches have been developed to ameliorate this problem.…
Wavelet (Besov) priors are a promising way of reconstructing indirectly measured fields in a regularized manner. We demonstrate how wavelets can be used as a localized basis for reconstructing permeability fields with sharp interfaces from…
We present a new approach to the electromagnetic inverse problem that explicitly addresses the ambiguity associated with its ill-posed character. Rather than calculating a single ``best'' solution according to some criterion, our approach…
Constrained learning is prevalent in many statistical tasks. Recent work proposes distance-to-set penalties to derive estimators under general constraints that can be specified as sets, but focuses on obtaining point estimates that do not…
We study empirical and hierarchical Bayes approaches to the problem of estimating an infinite-dimensional parameter in mildly ill-posed inverse problems. We consider a class of prior distributions indexed by a hyperparameter that quantifies…
From mostly a measure-theoretic consideration, we show that for every nonnegative, finite, and $L^{1}$ function on a given finite measure space there is some nontrivial sequence of real numbers such that the series, obtained from summing…
Diffusion models (DMs) have proven to be effective in modeling high-dimensional distributions, leading to their widespread adoption for representing complex priors in Bayesian inverse problems (BIPs). However, current DM-based posterior…
Benchmark datasets for visual recognition assume that data is uniformly distributed, while real-world datasets obey long-tailed distribution. Current approaches handle the long-tailed problem to transform the long-tailed dataset to uniform…
Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based…
In this paper we first provide several conditional limit theorems for L\'evy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of…
We consider the passage time problem for L\'evy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case),…
Bayesian hierarchical models can provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models typically comprise a conditionally Gaussian prior model for the unknown which is augmented by a generalized…
In inverse problems, the parameters of a model are estimated based on observations of the model response. The Bayesian approach is powerful for solving such problems; one formulates a prior distribution for the parameter state that is…
We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely…
Accurately quantifying tail risks-rare but high-impact events such as financial crashes or extreme weather-is a central challenge in risk management, with serially dependent data. We develop a Bayesian framework based on the Generalized…
We study non-linear Bayesian inverse problems arising from semilinear partial differential equations (PDEs) that can be transformed into linear Bayesian inverse problems. We are then able to extend the early stopping for Ensemble…