Uncertainty quantification (UQ) in scientific machine learning (SciML) combines the powerful predictive power of SciML with methods for quantifying the reliability of the learned models. However, two major challenges remain: limited interpretability and expensive training procedures. We provide a new interpretation for UQ problems by establishing a new theoretical connection between some Bayesian inference problems arising in SciML and viscous Hamilton-Jacobi partial differential equations (HJ PDEs). Namely, we show that the posterior mean and covariance can be recovered from the spatial gradient and Hessian of the solution to a viscous HJ PDE. As a first exploration of this connection, we specialize to Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors. In this case, the associated viscous HJ PDEs can be solved using Riccati ODEs, and we develop a new Riccati-based methodology that provides computational advantages when continuously updating the model predictions. Specifically, our Riccati-based approach can efficiently add or remove data points to the training set invariant to the order of the data and continuously tune hyperparameters. Moreover, neither update requires retraining on or access to previously incorporated data. We provide several examples from SciML involving noisy data and \textit{epistemic uncertainty} to illustrate the potential advantages of our approach. In particular, this approach's amenability to data streaming applications demonstrates its potential for real-time inferences, which, in turn, allows for applications in which the predicted uncertainty is used to dynamically alter the learning process.

In this paper, we investigate Bayesian model compression in federated learning (FL) to construct sparse models that can achieve both communication and computation efficiencies. We propose a decentralized Turbo variational Bayesian inference (D-Turbo-VBI) FL framework where we firstly propose a hierarchical sparse prior to promote a clustered sparse structure in the weight matrix. Then, by carefully integrating message passing and VBI with a decentralized turbo framework, we propose the D-Turbo-VBI algorithm which can (i) reduce both upstream and downstream communication overhead during federated training, and (ii) reduce the computational complexity during local inference. Additionally, we establish the convergence property for thr proposed D-Turbo-VBI algorithm. Simulation results show the significant gain of our proposed algorithm over the baselines in reducing communication overhead during federated training and computational complexity of final model.

With the advancement in generative language models, the selection of prompts has gained significant attention in recent years. A prompt is an instruction or description provided by the user, serving as a guide for the generative language model in content generation. Despite existing methods for prompt selection that are based on human labor, we consider facilitating this selection through simulation optimization, aiming to maximize a pre-defined score for the selected prompt. Specifically, we propose a two-stage framework. In the first stage, we determine a feasible set of prompts in sufficient numbers, where each prompt is represented by a moderate-dimensional vector. In the subsequent stage for evaluation and selection, we construct a surrogate model of the score regarding the moderate-dimensional vectors that represent the prompts. We propose sequentially selecting the prompt for evaluation based on this constructed surrogate model. We prove the consistency of the sequential evaluation procedure in our framework. We also conduct numerical experiments to demonstrate the efficacy of our proposed framework, providing practical instructions for implementation.

We show that established performance metrics in binary classification, such as the F-score, the Jaccard similarity coefficient or Matthews' correlation coefficient (MCC), are not robust to class imbalance in the sense that if the proportion of the minority class tends to $0$, the true positive rate (TPR) of the Bayes classifier under these metrics tends to $0$ as well. Thus, in imbalanced classification problems, these metrics favour classifiers which ignore the minority class. To alleviate this issue we introduce robust modifications of the F-score and the MCC for which, even in strongly imbalanced settings, the TPR is bounded away from $0$. We numerically illustrate the behaviour of the various performance metrics in simulations as well as on a credit default data set. We also discuss connections to the ROC and precision-recall curves and give recommendations on how to combine their usage with performance metrics.

We study the problem of online sequential decision-making given auxiliary demonstrations from experts who made their decisions based on unobserved contextual information. These demonstrations can be viewed as solving related but slightly different tasks than what the learner faces. This setting arises in many application domains, such as self-driving cars, healthcare, and finance, where expert demonstrations are made using contextual information, which is not recorded in the data available to the learning agent. We model the problem as a zero-shot meta-reinforcement learning setting with an unknown task distribution and a Bayesian regret minimization objective, where the unobserved tasks are encoded as parameters with an unknown prior. We propose the Experts-as-Priors algorithm (ExPerior), a non-parametric empirical Bayes approach that utilizes the principle of maximum entropy to establish an informative prior over the learner's decision-making problem. This prior enables the application of any Bayesian approach for online decision-making, such as posterior sampling. We demonstrate that our strategy surpasses existing behaviour cloning and online algorithms for multi-armed bandits and reinforcement learning, showcasing the utility of our approach in leveraging expert demonstrations across different decision-making setups.

This work presents a framework for automatically extracting physical object properties, such as material composition, mass, volume, and stiffness, through robot manipulation and a database of object measurements. The framework involves exploratory action selection to maximize learning about objects on a table. A Bayesian network models conditional dependencies between object properties, incorporating prior probability distributions and uncertainty associated with measurement actions. The algorithm selects optimal exploratory actions based on expected information gain and updates object properties through Bayesian inference. Experimental evaluation demonstrates effective action selection compared to a baseline and correct termination of the experiments if there is nothing more to be learned. The algorithm proved to behave intelligently when presented with trick objects with material properties in conflict with their appearance. The robot pipeline integrates with a logging module and an online database of objects, containing over 24,000 measurements of 63 objects with different grippers. All code and data are publicly available, facilitating automatic digitization of objects and their physical properties through exploratory manipulations.

Preference modelling lies at the intersection of economics, decision theory, machine learning and statistics. By understanding individuals' preferences and how they make choices, we can build products that closely match their expectations, paving the way for more efficient and personalised applications across a wide range of domains. The objective of this tutorial is to present a cohesive and comprehensive framework for preference learning with Gaussian Processes (GPs), demonstrating how to seamlessly incorporate rationality principles (from economics and decision theory) into the learning process. By suitably tailoring the likelihood function, this framework enables the construction of preference learning models that encompass random utility models, limits of discernment, and scenarios with multiple conflicting utilities for both object- and label-preference. This tutorial builds upon established research while simultaneously introducing some novel GP-based models to address specific gaps in the existing literature.

The von Mises-Fisher distribution as an exponential family can be expressed in terms of either its natural or its mean parameters. Unfortunately, however, the normalization function for the distribution in terms of its mean parameters is not available in closed form, limiting the practicality of the mean parametrization and complicating maximum-likelihood estimation more generally. We derive a second-order ordinary differential equation, the solution to which yields the mean-parameter normalizer along with its first two derivatives, as well as the variance function of the family. We also provide closed-form approximations to the solution of the differential equation. This allows rapid evaluation of both densities and natural parameters in terms of mean parameters. We show applications to topic modeling with mixtures of von Mises-Fisher distributions using Bregman Clustering.

Causality reveals fundamental principles behind data distributions in real-world scenarios, and the capability of large language models (LLMs) to understand causality directly impacts their efficacy across explaining outputs, adapting to new evidence, and generating counterfactuals. With the proliferation of LLMs, the evaluation of this capacity is increasingly garnering attention. However, the absence of a comprehensive benchmark has rendered existing evaluation studies being straightforward, undiversified, and homogeneous. To address these challenges, this paper proposes a comprehensive benchmark, namely CausalBench, to evaluate the causality understanding capabilities of LLMs. Originating from the causal research community, CausalBench encompasses three causal learning-related tasks, which facilitate a convenient comparison of LLMs' performance with classic causal learning algorithms. Meanwhile, causal networks of varying scales and densities are integrated in CausalBench, to explore the upper limits of LLMs' capabilities across task scenarios of varying difficulty. Notably, background knowledge and structured data are also incorporated into CausalBench to thoroughly unlock the underlying potential of LLMs for long-text comprehension and prior information utilization. Based on CausalBench, this paper evaluates nineteen leading LLMs and unveils insightful conclusions in diverse aspects. Firstly, we present the strengths and weaknesses of LLMs and quantitatively explore the upper limits of their capabilities across various scenarios. Meanwhile, we further discern the adaptability and abilities of LLMs to specific structural networks and complex chain of thought structures. Moreover, this paper quantitatively presents the differences across diverse information sources and uncovers the gap between LLMs' capabilities in causal understanding within textual contexts and numerical domains.

Unsupervised learning is a training approach in the situation where ground truth data is unavailable, such as inverse imaging problems. We present an unsupervised Bayesian training approach to learning convex neural network regularizers using a fixed noisy dataset, based on a dual Markov chain estimation method. Compared to classical supervised adversarial regularization methods, where there is access to both clean images as well as unlimited to noisy copies, we demonstrate close performance on natural image Gaussian deconvolution and Poisson denoising tasks.