The special role of epistemic uncertainty in Machine Learning is now well recognised, and an increasing amount of research is focused on methods for dealing specifically with such a lack of knowledge. Yet, most often, a probabilistic representation is considered for both aleatoric and epistemic uncertainties, hence creating challenges in applications where decoupling these two types of uncertainty is necessary. In this work, we show that an alternative representation of epistemic uncertainty, based on possibility theory, maintains many of the convenient features of standard Bayesian inference while displaying specific behaviours and properties that closely match the ones of an intuitive notion of information. Our main contributions are: i) a general framework for jointly representing epistemic and aleatoric uncertainties, ii) a Bernstein-von Mises theorem for the analogue of Bayes’ rule in possibility theory, iii) a version of the law of large numbers and of the central limit theorem for the associated variables, and iv) an analysis of the properties of the possibilistic maximum a posteriori. These results highlight that a dedicated and principled representation of epistemic uncertainty, that is compatible with standard Bayesian inference and preserves many of its strengths, is attainable.

We propose a scalable kinetic Langevin dynamics algorithm for sampling parameter spaces of big data and AI applications. Our scheme combines a symmetric forward/backward sweep over minibatches with a symmetric discretization of Langevin dynamics. For a particular Langevin splitting method (UBU), we show that the resulting Symmetric Minibatch Splitting-UBU (SMS-UBU) integrator has bias $\mathcal{O}(h^2 d^{1/2})$ in dimension $d>0$ with stepsize $h>0$, despite only using one minibatch per iteration, thus providing excellent control of the sampling bias as a function of the stepsize. We apply the algorithm to explore local modes of the posterior distribution of Bayesian neural networks (BNNs) and evaluate the calibration performance of the posterior predictive probabilities for neural networks with convolutional neural network architectures for classification problems on three different datasets (Fashion-MNIST, Celeb-A and chest X-ray). Our results indicate that BNNs sampled with SMS-UBU can offer significantly better calibration performance compared to standard methods of training and stochastic weight averaging.

Kernels effectively represent nonlocal dependencies and are extensively employed in formu- lating operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse prob- lem is often severely ill-posed with a data-dependent normal operator. Traditional Bayesian methods address the ill-posedness by a non-degenerate prior, which may result in an unsta- ble posterior mean in the small noise regime, especially when data induces a perturbation in the null space of the normal operator. We propose a new data-adaptive Reproducing Kernel Hilbert Space (RKHS) prior, which ensures the stability of the posterior mean in the small noise regime. We analyze this adaptive prior and showcase its efficacy through applications on Toeplitz matrices and integral operators. Numerical experiments reveal that fixed non-degenerate priors can produce divergent posterior means under errors from discretization, model inaccuracies, partial observations, or erroneous noise assumptions. In contrast, our data-adaptive RKHS prior consistently yields convergent posterior means.