We introduce a novel deep operator network (DeepONet) framework that incorporates generalised variational inference (GVI) using R\'enyi's $\alpha$-divergence to learn complex operators while quantifying uncertainty. By incorporating Bayesian neural networks as the building blocks for the branch and trunk networks, our framework endows DeepONet with uncertainty quantification. The use of R\'enyi's $\alpha$-divergence, instead of the Kullback-Leibler divergence (KLD), commonly used in standard variational inference, mitigates issues related to prior misspecification that are prevalent in Variational Bayesian DeepONets. This approach offers enhanced flexibility and robustness. We demonstrate that modifying the variational objective function yields superior results in terms of minimising the mean squared error and improving the negative log-likelihood on the test set. Our framework's efficacy is validated across various mechanical systems, where it outperforms both deterministic and standard KLD-based VI DeepONets in predictive accuracy and uncertainty quantification. The hyperparameter $\alpha$, which controls the degree of robustness, can be tuned to optimise performance for specific problems. We apply this approach to a range of mechanics problems, including gravity pendulum, advection-diffusion, and diffusion-reaction systems. Our findings underscore the potential of $\alpha$-VI DeepONet to advance the field of data-driven operator learning and its applications in engineering and scientific domains.

Data for several applications in diverse fields can be represented as multiple matrices that are linked across rows or columns. This is particularly common in molecular biomedical research, in which multiple molecular "omics" technologies may capture different feature sets (e.g., corresponding to rows in a matrix) and/or different sample populations (corresponding to columns). This has motivated a large body of work on integrative matrix factorization approaches that identify and decompose low-dimensional signal that is shared across multiple matrices or specific to a given matrix. We propose an empirical variational Bayesian approach to this problem that has several advantages over existing techniques, including the flexibility to accommodate shared signal over any number of row or column sets (i.e., bidimensional integration), an intuitive model-based objective function that yields appropriate shrinkage for the inferred signals, and a relatively efficient estimation algorithm with no tuning parameters. A general result establishes conditions for the uniqueness of the underlying decomposition for a broad family of methods that includes the proposed approach. For scenarios with missing data, we describe an associated iterative imputation approach that is novel for the single-matrix context and a powerful approach for "blockwise" imputation (in which an entire row or column is missing) in various linked matrix contexts. Extensive simulations show that the method performs very well under different scenarios with respect to recovering underlying low-rank signal, accurately decomposing shared and specific signals, and accurately imputing missing data. The approach is applied to gene expression and miRNA data from breast cancer tissue and normal breast tissue, for which it gives an informative decomposition of variation and outperforms alternative strategies for missing data imputation.

Gaussian processes (GPs) are widely used in non-parametric Bayesian modeling, and play an important role in various statistical and machine learning applications. In a variety tasks of uncertainty quantification, generating random sample paths of GPs is of interest. As GP sampling requires generating high-dimensional Gaussian random vectors, it is computationally challenging if a direct method, such as the Cholesky decomposition, is used. In this paper, we propose a scalable algorithm for sampling random realizations of the prior and posterior of GP models. The proposed algorithm leverages inducing points approximation with sparse grids, as well as additive Schwarz preconditioners, which reduce computational complexity, and ensure fast convergence. We demonstrate the efficacy and accuracy of the proposed method through a series of experiments and comparisons with other recent works.

PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an It\^o equation from a stochastic dissipative Hamiltonian dynamical system, acting as the MCMC generator, for which the KDE-estimated probability measure with the training dataset is the invariant measure. PLoM performs a projection on a reduced-order vector basis related to the training dataset, using the diffusion maps (DMAPS) basis constructed with a time-independent isotropic kernel. In this paper, we propose a new ISDE projection vector basis built from a transient anisotropic kernel, providing an alternative to the DMAPS basis to improve statistical surrogates for stochastic manifolds with heterogeneous data. The construction ensures that for times near the initial time, the DMAPS basis coincides with the transient basis. For larger times, the differences between the two bases are characterized by the angle of their spanned vector subspaces. The optimal instant yielding the optimal transient basis is determined using an estimation of mutual information from Information Theory, which is normalized by the entropy estimation to account for the effects of the number of realizations used in the estimations. Consequently, this new vector basis better represents statistical dependencies in the learned probability measure for any dimension. Three applications with varying levels of statistical complexity and data heterogeneity validate the proposed theory, showing that the transient anisotropic kernel improves the learned probability measure.

While fiducial inference was widely considered a big blunder by R.A. Fisher, the goal he initially set --`inferring the uncertainty of model parameters on the basis of observations' -- has been continually pursued by many statisticians. To this end, we develop a new statistical inference method called extended Fiducial inference (EFI). The new method achieves the goal of fiducial inference by leveraging advanced statistical computing techniques while remaining scalable for big data. EFI involves jointly imputing random errors realized in observations using stochastic gradient Markov chain Monte Carlo and estimating the inverse function using a sparse deep neural network (DNN). The consistency of the sparse DNN estimator ensures that the uncertainty embedded in observations is properly propagated to model parameters through the estimated inverse function, thereby validating downstream statistical inference. Compared to frequentist and Bayesian methods, EFI offers significant advantages in parameter estimation and hypothesis testing. Specifically, EFI provides higher fidelity in parameter estimation, especially when outliers are present in the observations; and eliminates the need for theoretical reference distributions in hypothesis testing, thereby automating the statistical inference process. EFI also provides an innovative framework for semi-supervised learning.

Sequential Monte Carlo (SMC) methods are powerful tools for Bayesian inference but suffer from requiring many particles for accurate estimates, leading to high computational costs. We introduce persistent sampling (PS), an extension of SMC that mitigates this issue by allowing particles from previous iterations to persist. This generates a growing, weighted ensemble of particles distributed across iterations. In each iteration, PS utilizes multiple importance sampling and resampling from the mixture of all previous distributions to produce the next generation of particles. This addresses particle impoverishment and mode collapse, resulting in more accurate posterior approximations. Furthermore, this approach provides lower-variance marginal likelihood estimates for model comparison. Additionally, the persistent particles improve transition kernel adaptation for efficient exploration. Experiments on complex distributions show that PS consistently outperforms standard methods, achieving lower squared bias in posterior moment estimation and significantly reduced marginal likelihood errors, all at a lower computational cost. PS offers a robust, efficient, and scalable framework for Bayesian inference.

eBay partners with external vendors, which allows customers to freely select a vendor to complete their eBay experiences. However, vendor outages can hinder customer experiences. Consequently, eBay can disable a problematic vendor to prevent customer loss. Disabling the vendor too late risks losing customers willing to switch to other vendors, while disabling it too early risks losing those unwilling to switch. In this paper, we propose a data-driven solution to answer whether eBay should disable a problematic vendor and when to disable it. Our solution involves forecasting customer behavior. First, we use a multiplicative seasonality model to represent behavior if all vendors are fully functioning. Next, we use a Monte Carlo simulation to represent behavior if the problematic vendor remains enabled. Finally, we use a linear model to represent behavior if the vendor is disabled. By comparing these forecasts, we determine the optimal time for eBay to disable the problematic vendor.

Bayesian Inference with Markov Chain Monte Carlo requires efficient computation of the likelihood function. In some scientific applications, the likelihood must be computed by numerically solving a partial differential equation, which can be prohibitively expensive. We demonstrate that some such problems can be made tractable by amortizing the computation with a surrogate likelihood function implemented by a neural network. We show that this has two additional benefits: reducing noise in the likelihood evaluations and providing fast gradient calculations. In experiments, the approach is applied to a model of heliospheric transport of galactic cosmic rays, where it enables efficient sampling from the posterior of latent parameters in the Parker equation.

We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive distributions with a pre-specified probability. Furthermore, we propose, Trust-Bayes, a novel optimization framework for Bayesian meta learning which is cognizant of trustworthy uncertainty quantification without explicit assumptions on the prior model/distribution of the functions. We characterize the lower bounds of the probabilities of the ground truth being captured by the specified intervals and analyze the sample complexity with respect to the feasible probability for trustworthy uncertainty quantification. Monte Carlo simulation of a case study using Gaussian process regression is conducted for verification and comparison with the Meta-prior algorithm.

This paper describes a discrete state-space model -- and accompanying methods -- for generative modelling. This model generalises partially observed Markov decision processes to include paths as latent variables, rendering it suitable for active inference and learning in a dynamic setting. Specifically, we consider deep or hierarchical forms using the renormalisation group. The ensuing renormalising generative models (RGM) can be regarded as discrete homologues of deep convolutional neural networks or continuous state-space models in generalised coordinates of motion. By construction, these scale-invariant models can be used to learn compositionality over space and time, furnishing models of paths or orbits; i.e., events of increasing temporal depth and itinerancy. This technical note illustrates the automatic discovery, learning and deployment of RGMs using a series of applications. We start with image classification and then consider the compression and generation of movies and music. Finally, we apply the same variational principles to the learning of Atari-like games.