Related papers: Scale invariant process regression: Towards Bayesi…

Bayesian methods are actively used for parameter identification and uncertainty quantification when solving nonlinear inverse problems with random noise. However, there are only few theoretical results justifying the Bayesian approach.…

Bayesian inference with computationally expensive likelihood evaluations remains a significant challenge in many scientific domains. We propose normalizing flow regression (NFR), a novel offline inference method for approximating posterior…

It is by now well-established that modern over-parameterized models seem to elude the bias-variance tradeoff and generalize well despite overfitting noise. Many recent works attempt to analyze this phenomenon in the relatively tractable…

Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(\theta): \theta \in \Theta\}$ and on Gaussian process…

Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns…

Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i.e. BNNs assign different posterior densities to different parametrizations of…

This paper presents a detailed theoretical analysis of the three stochastic approximation proximal gradient algorithms proposed in our companion paper [49] to set regularization parameters by marginal maximum likelihood estimation. We prove…

We consider the statistical inverse problem of recovering a function $f: M \to \mathbb R$, where $M$ is a smooth compact Riemannian manifold with boundary, from measurements of general $X$-ray transforms $I_a(f)$ of $f$, corrupted by…

Bayesian inference provides a flexible way of combining data with prior information. However, quantile regression is not equipped with a parametric likelihood, and therefore, Bayesian inference for quantile regression demands careful…

Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many…

Bayesian methods in machine learning, such as Gaussian processes, have great advantages com-pared to other techniques. In particular, they provide estimates of the uncertainty associated with a prediction. Extending the Bayesian approach to…

In the Bayesian approach, the a priori knowledge about the input of a mathematical model is described via a probability measure. The joint distribution of the unknown input and the data is then conditioned, using Bayes' formula, giving rise…

The kernel function and its hyperparameters are the central model selection choice in a Gaussian proces (Rasmussen and Williams, 2006). Typically, the hyperparameters of the kernel are chosen by maximising the marginal likelihood, an…

We study Nystr\"om type subsampling approaches to large scale kernel methods, and prove learning bounds in the statistical learning setting, where random sampling and high probability estimates are considered. In particular, we prove that…

Learning to infer Bayesian posterior from a few-shot dataset is an important step towards robust meta-learning due to the model uncertainty inherent in the problem. In this paper, we propose a novel Bayesian model-agnostic meta-learning…

We consider joint inversion for two or more unknown parameters from observational data in the Bayesian framework. Standard approaches often either treat the parameters as independent or impose structural similarity through regularisation…

Majorization-minimization (MM) is a standard iterative optimization technique which consists in minimizing a sequence of convex surrogate functionals. MM approaches have been particularly successful to tackle inverse problems and…

Generalization beyond a training dataset is a main goal of machine learning, but theoretical understanding of generalization remains an open problem for many models. The need for a new theory is exacerbated by recent observations in deep…

In many environmental applications involving spatially-referenced data, limitations on the number and locations of observations motivate the need for practical and efficient models for spatial interpolation, or kriging. A key component of…

Inverse problems and, in particular, inferring unknown or latent parameters from data are ubiquitous in engineering simulations. A predominant viewpoint in identifying unknown parameters is Bayesian inference where both prior information…