We propose a physics-informed machine learning method for uncertainty quantification in high-dimensional inverse problems. In this method, the states and parameters of partial differential equations (PDEs) are approximated with truncated conditional Karhunen-Lo\`eve expansions (CKLEs), which, by construction, match the measurements of the respective variables. The maximum a posteriori (MAP) solution of the inverse problem is formulated as a minimization problem over CKLE coefficients where the loss function is the sum of the norm of PDE residuals and the $\ell_2$ regularization term. This MAP formulation is known as the physics-informed CKLE (PICKLE) method. Uncertainty in the inverse solution is quantified in terms of the posterior distribution of CKLE coefficients, and we sample the posterior by solving a randomized PICKLE minimization problem, formulated by adding zero-mean Gaussian perturbations in the PICKLE loss function. We call the proposed approach the randomized PICKLE (rPICKLE) method. For linear and low-dimensional nonlinear problems (15 CKLE parameters), we show analytically and through comparison with Hamiltonian Monte Carlo (HMC) that the rPICKLE posterior converges to the true posterior given by the Bayes rule. For high-dimensional non-linear problems with 2000 CKLE parameters, we numerically demonstrate that rPICKLE posteriors are highly informative--they provide mean estimates with an accuracy comparable to the estimates given by the MAP solution and the confidence interval that mostly covers the reference solution. We are not able to obtain the HMC posterior to validate rPICKLE's convergence to the true posterior due to the HMC's prohibitive computational cost for the considered high-dimensional problems. Our results demonstrate the advantages of rPICKLE over HMC for approximately sampling high-dimensional posterior distributions subject to physics constraints.

Reinforcement learning-based large language models, such as ChatGPT, are believed to have potential to aid human experts in many domains, including healthcare. There is, however, little work on ChatGPT's ability to perform a key task in healthcare: formal, probabilistic medical diagnostic reasoning. This type of reasoning is used, for example, to update a pre-test probability to a post-test probability. In this work, we probe ChatGPT's ability to perform this task. In particular, we ask ChatGPT to give examples of how to use Bayes rule for medical diagnosis. Our prompts range from queries that use terminology from pure probability (e.g., requests for a posterior of A given B and C) to queries that use terminology from medical diagnosis (e.g., requests for a posterior probability of Covid given a test result and cough). We show how the introduction of medical variable names leads to an increase in the number of errors that ChatGPT makes. Given our results, we also show how one can use prompt engineering to facilitate ChatGPT's partial avoidance of these errors. We discuss our results in light of recent commentaries on sensitivity and specificity. We also discuss how our results might inform new research directions for large language models.

In nonstationary bandit learning problems, the decision-maker must continually gather information and adapt their action selection as the latent state of the environment evolves. In each time period, some latent optimal action maximizes expected reward under the environment state. We view the optimal action sequence as a stochastic process, and take an information-theoretic approach to analyze attainable performance. We bound limiting per-period regret in terms of the entropy rate of the optimal action process. The bound applies to a wide array of problems studied in the literature and reflects the problem's information structure through its information-ratio.

In pool-based active learning, the learner is given an unlabeled data set and aims to efficiently learn the unknown hypothesis by querying the labels of the data points. This can be formulated as the classical Optimal Decision Tree (ODT) problem: Given a set of tests, a set of hypotheses, and an outcome for each pair of test and hypothesis, our objective is to find a low-cost testing procedure (i.e., decision tree) that identifies the true hypothesis. This optimization problem has been extensively studied under the assumption that each test generates a deterministic outcome. However, in numerous applications, for example, clinical trials, the outcomes may be uncertain, which renders the ideas from the deterministic setting invalid. In this work, we study a fundamental variant of the ODT problem in which some test outcomes are noisy, even in the more general case where the noise is persistent, i.e., repeating a test gives the same noisy output. Our approximation algorithms provide guarantees that are nearly best possible and hold for the general case of a large number of noisy outcomes per test or per hypothesis where the performance degrades continuously with this number. We numerically evaluated our algorithms for identifying toxic chemicals and learning linear classifiers, and observed that our algorithms have costs very close to the information-theoretic minimum.

Deep Neural Networks (DNNs) are powerful tools for various computer vision tasks, yet they often struggle with reliable uncertainty quantification - a critical requirement for real-world applications. Bayesian Neural Networks (BNN) are equipped for uncertainty estimation but cannot scale to large DNNs that are highly unstable to train. To address this challenge, we introduce the Adaptable Bayesian Neural Network (ABNN), a simple and scalable strategy to seamlessly transform DNNs into BNNs in a post-hoc manner with minimal computational and training overheads. ABNN preserves the main predictive properties of DNNs while enhancing their uncertainty quantification abilities through simple BNN adaptation layers (attached to normalization layers) and a few fine-tuning steps on pre-trained models. We conduct extensive experiments across multiple datasets for image classification and semantic segmentation tasks, and our results demonstrate that ABNN achieves state-of-the-art performance without the computational budget typically associated with ensemble methods.

Reinforcement learning from human feedback (RLHF) is a variant of reinforcement learning (RL) that learns from human feedback instead of relying on an engineered reward function. Building on prior work on the related setting of preference-based reinforcement learning (PbRL), it stands at the intersection of artificial intelligence and human-computer interaction. This positioning offers a promising avenue to enhance the performance and adaptability of intelligent systems while also improving the alignment of their objectives with human values. The training of Large Language Models (LLMs) has impressively demonstrated this potential in recent years, where RLHF played a decisive role in targeting the model's capabilities toward human objectives. This article provides a comprehensive overview of the fundamentals of RLHF, exploring the intricate dynamics between machine agents and human input. While recent focus has been on RLHF for LLMs, our survey adopts a broader perspective, examining the diverse applications and wide-ranging impact of the technique. We delve into the core principles that underpin RLHF, shedding light on the symbiotic relationship between algorithms and human feedback, and discuss the main research trends in the field. By synthesizing the current landscape of RLHF research, this article aims to provide researchers as well as practitioners with a comprehensive understanding of this rapidly growing field of research.

One of the grand challenges of cell biology is inferring the gene regulatory network (GRN) which describes interactions between genes and their products that control gene expression and cellular function. We can treat this as a causal discovery problem but with two non-standard challenges: (1) regulatory networks are inherently cyclic so we should not model a GRN as a directed acyclic graph (DAG), and (2) observations have significant measurement noise, so for typical sample sizes there will always be a large equivalence class of graphs that are likely given the data, and we want methods that capture this uncertainty. Existing methods either focus on challenge (1), identifying cyclic structure from dynamics, or on challenge (2) learning complex Bayesian posteriors over DAGs, but not both. In this paper we leverage the fact that it is possible to estimate the "velocity" of gene expression with RNA velocity techniques to develop an approach that addresses both challenges. Because we have access to velocity information, we can treat the Bayesian structure learning problem as a problem of sparse identification of a dynamical system, capturing cyclic feedback loops through time. Since our objective is to model uncertainty over discrete structures, we leverage Generative Flow Networks (GFlowNets) to estimate the posterior distribution over the combinatorial space of possible sparse dependencies. Our results indicate that our method learns posteriors that better encapsulate the distributions of cyclic structures compared to counterpart state-of-the-art Bayesian structure learning approaches.

We propose a novel Deep Reinforcement Learning (DRL) architecture for sequential decision processes under uncertainty, as encountered in inspection and maintenance (I&M) planning. Unlike other DRL algorithms for (I&M) planning, the proposed +RQN architecture dispenses with computing the belief state and directly handles erroneous observations instead. We apply the algorithm to a basic I&M planning problem for a one-component system subject to deterioration. In addition, we investigate the performance of Monte Carlo tree search for the I&M problem and compare it to the +RQN. The comparison includes a statistical analysis of the two methods' resulting policies, as well as their visualization in the belief space.

Although the foundations of ranking are well established, the ranking literature has primarily been focused on simple, unimodal models, e.g. the Mallows and Plackett-Luce models, that define distributions centered around a single total ordering. Explicit mixture models have provided some tools for modelling multimodal ranking data, though learning such models from data is often difficult. In this work, we contribute a contextual repeated selection (CRS) model that leverages recent advances in choice modeling to bring a natural multimodality and richness to the rankings space. We provide rigorous theoretical guarantees for maximum likelihood estimation under the model through structure-dependent tail risk and expected risk bounds. As a by-product, we also furnish the first tight bounds on the expected risk of maximum likelihood estimators for the multinomial logit (MNL) choice model and the Plackett-Luce (PL) ranking model, as well as the first tail risk bound on the PL ranking model. The CRS model significantly outperforms existing methods for modeling real world ranking data in a variety of settings, from racing to rank choice voting.

In recent years, there has been a significant transformation in medical research methodologies towards the adoption of Deep Learning (DL) techniques for predicting critical events, such as disease development and patient mortality. Despite their potential to handle complex data, practical applications in this domain remain limited, with most studies still relying on traditional statistical methods. Survival Analysis (SA), or time-to-event analysis, is an essential tool for studying specific events in various disciplines, not only in medicine but also in fields such as recommendation systems [1], employee retention [2], market modeling [3], and financial risk assessment [4]. According to the existing literature, the Cox proportional hazards model (Cox-PH) [5] is the dominant SA method that offers a semiparametric regression solution to the non-parametric Kaplan-Meier estimator problem [6]. Unlike the Kaplan-Meier method, which uses a single covariate, Cox-PH incorporates multiple covariates to predict event times and assess their impact on the hazard rate at specific time points. However, it is crucial to acknowledge that the Cox-PH model is built on certain strong assumptions. One of these is the proportional hazards assumption, which posits that different individuals have hazard functions that remain constant over time. Furthermore, the model assumes a linear relation between the natural logarithm of the relative hazard (the ratio of the hazard at time t to the baseline hazard) and the covariates. Furthermore, it assumes the absence of interactions among these covariates.