We propose a novel approach for solving inverse-problems with high-dimensional inputs and an expensive forward mapping. It leverages joint deep generative modelling to transfer the original problem spaces to a lower dimensional latent space. By jointly modelling input and output variables and endowing the latent with a prior distribution, the fitted probabilistic model indirectly gives access to the approximate conditional distributions of interest. Since model error and observational noise with unknown distributions are common in practice, we resort to likelihood-free inference with Approximate Bayesian Computation (ABC). Our method calls on ABC by Subset Simulation to explore the regions of the latent space with dissimilarities between generated and observed outputs below prescribed thresholds. We diagnose the diversity of approximate posterior solutions by monitoring the probability content of these regions as a function of the threshold. We further analyze the curvature of the resulting diagnostic curve to propose an adequate ABC threshold. When applied to a cross-borehole tomography example from geophysics, our approach delivers promising performance without using prior knowledge of the forward nor of the noise distribution.

Generalised Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and can therefore be used to confer robustness against possible misspecification of the likelihood. Here we consider generalised Bayesian inference with a Stein discrepancy as a loss function, motivated by applications in which the likelihood contains an intractable normalisation constant. In this context, the Stein discrepancy circumvents evaluation of the normalisation constant and produces generalised posteriors that are either closed form or accessible using standard Markov chain Monte Carlo. On a theoretical level, we show consistency, asymptotic normality, and bias-robustness of the generalised posterior, highlighting how these properties are impacted by the choice of Stein discrepancy. Then, we provide numerical experiments on a range of intractable distributions, including applications to kernel-based exponential family models and non-Gaussian graphical models.

Topological data analysis (TDA) studies the shape patterns of data. Persistent homology (PH) is a widely used method in TDA that summarizes homological features of data at multiple scales and stores this in persistence diagrams (PDs). As TDA is commonly used in the analysis of high dimensional data sets, a sufficiently large amount of PDs that allow performing statistical analysis is typically unavailable or requires inordinate computational resources. In this paper, we propose random persistence diagram generation (RPDG), a method that generates a sequence of random PDs from the ones produced by the data. RPDG is underpinned (i) by a parametric model based on pairwise interacting point processes for inference of persistence diagrams and (ii) by a reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithm for generating samples of PDs. The parametric model combines a Dirichlet partition to capture spatial homogeneity of the location of points in a PD and a step function to capture the pairwise interaction between them. The RJ-MCMC algorithm incorporates trans-dimensional addition and removal of points and same-dimensional relocation of points across samples of PDs. The efficacy of RPDG is demonstrated via an example and a detailed comparison with other existing methods is presented.

There is significant need for principled uncertainty reasoning in machine learning systems as they are increasingly deployed in safety-critical domains. A new approach with uncertainty-aware neural networks (NNs), based on learning evidential distributions for aleatoric and epistemic uncertainties, shows promise over traditional deterministic methods and typical Bayesian NNs, yet several important gaps in the theory and implementation of these networks remain. We discuss three issues with a proposed solution to extract aleatoric and epistemic uncertainties from regression-based neural networks. The approach derives a technique by placing evidential priors over the original Gaussian likelihood function and training the NN to infer the hyperparameters of the evidential distribution. Doing so allows for the simultaneous extraction of both uncertainties without sampling or utilization of out-of-distribution data for univariate regression tasks. We describe the outstanding issues in detail, provide a possible solution, and generalize the deep evidential regression technique for multivariate cases.

Learning exists in the context of data, yet notions of confidence typically focus on model predictions, not label quality. Confident learning (CL) is an alternative approach which focuses instead on label quality by characterizing and identifying label errors in datasets, based on the principles of pruning noisy data, counting with probabilistic thresholds to estimate noise, and ranking examples to train with confidence. Whereas numerous studies have developed these principles independently, here, we combine them, building on the assumption of a class-conditional noise process to directly estimate the joint distribution between noisy (given) labels and uncorrupted (unknown) labels. This results in a generalized CL which is provably consistent and experimentally performant. We present sufficient conditions where CL exactly finds label errors, and show CL performance exceeding seven recent competitive approaches for learning with noisy labels on the CIFAR dataset. Uniquely, the CL framework is not coupled to a specific data modality or model (e.g., we use CL to find several label errors in the presumed error-free MNIST dataset and improve sentiment classification on text data in Amazon Reviews). We also employ CL on ImageNet to quantify ontological class overlap (e.g., estimating 645 missile images are mislabeled as their parent class projectile), and moderately increase model accuracy (e.g., for ResNet) by cleaning data prior to training. These results are replicable using the open-source cleanlab release.

We consider the problem of forecasting the daily number of hospitalized COVID-19 patients at a single hospital site, in order to help administrators with logistics and planning. We develop several candidate hierarchical Bayesian models which directly capture the count nature of data via a generalized Poisson likelihood, model time-series dependencies via autoregressive and Gaussian process latent processes, and can share statistical strength across related sites. We demonstrate our approach on public datasets for 8 hospitals in Massachusetts, U.S.A. and 10 hospitals in the United Kingdom. Further prospective evaluation compares our approach favorably to baselines currently used by stakeholders at 3 related hospitals to forecast 2-week-ahead demand by rescaling state-level forecasts. The COVID-19 pandemic has created unprecedented demand for limited hospital resources across the globe. Emergency resource allocation decisions made by hospital administrators (such as planning additional personnel or provisioning beds and equipment) are crucial for achieving successful patient outcomes and avoiding overwhelmed capacity. However, at present hospitals often lack the ability to forecast what will be needed at their site in coming weeks. This may be especially true in under-resourced hospitals, due to constraints on funding, staff time and expertise, and other issues.

Fully convolutional U-shaped neural networks have largely been the dominant approach for pixel-wise image segmentation. In this work, we tackle two defects that hinder their deployment in real-world applications: 1) Predictions lack uncertainty quantification that may be crucial to many decision making systems; 2) Large memory storage and computational consumption demanding extensive hardware resources. To address these issues and improve their practicality we demonstrate a few-parameter compact Bayesian convolutional architecture, that achieves a marginal improvement in accuracy in comparison to related work using significantly fewer parameters and compute operations. The architecture combines parameter-efficient operations such as separable convolutions, bi-linear interpolation, multi-scale feature propagation and Bayesian inference for per-pixel uncertainty quantification through Monte Carlo Dropout. The best performing configurations required fewer than 2.5 million parameters on diverse challenging datasets with few observations.

Probability theory is far from being the most general mathematical theory of uncertainty. A number of arguments point at its inability to describe second-order ('Knightian') uncertainty. In response, a wide array of theories of uncertainty have been proposed, many of them generalisations of classical probability. As we show here, such frameworks can be organised into clusters sharing a common rationale, exhibit complex links, and are characterised by different levels of generality. Our goal is a critical appraisal of the current landscape in uncertainty theory.

Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data from neighboring surveys. To arrive at this result, we train the NF on pairs of low- and high-fidelity migrated images. In our numerical example, we obtain high-fidelity images from the Parihaka dataset and low-fidelity images are derived from these images through the process of demigration, followed by adding noise and migration. During inference, given shot records from a new neighboring seismic survey, we first compute the reverse-time migration image. Next, by feeding this low-fidelity migrated image to the NF we gain access to samples from the posterior distribution virtually for free. We use these samples to compute a high-fidelity image including a first assessment of the image's reliability. To our knowledge, this is the first attempt to train a conditional network on what we know from neighboring images to improve the current image and assess its reliability.

Emerging neuroscience technologies such as electrophysiology and calcium imaging can record from tens-of-thousands of neurons in the live animal brain while the animal is responding to stimuli and behaving freely. Scientists often seek to understand how neurons are communicating during certain stimuli or activities, something termed functional neural connectivity. To learn functional connections from large-scale neuroscience data, many have proposed using probabilistic graphical models (Yatsenko et al. 2015; Narayan et al. 2015; Chang et al. 2019), where each edge denotes conditional dependencies between nodes. Yet, applying such models in neuroscience poses a major challenge as only a small subset of neurons in the animal brain can be recorded at once, leading to abundant latent variables. Chandrasekaran et al. (2012) termed this the latent variable graphical model problem and proposed a convex program to solve this. While conceptually attractive, this approach poses several statistical, computational and practical challenges, discussed subsequently, for the task of learning functional neural connectivity from large-scale neuroscience data. Because of this, we are motivated to consider an incredibly simple solution to the latent variable graphical model problem: apply a hard thresholding operator to existing graph selection estimators. In this paper, we study this approach showing that thresholding has more desirable theoretical properties as well as superior empirical performance.