This textbook gives a systematic treatment of statistical machine learning for astronomical research, based on Bayesian inference. It shows how modern data analysis techniques connect to classical statistical methods and follow from familiar statistical principles. The Bayesian perspective, used throughout, gives priority to uncertainty quantification and statistical rigor, which scientific inference in astronomy needs. The textbook progresses from probability theory and Bayesian inference through supervised learning, including linear regression with measurement uncertainties, logistic regression and classification. Unsupervised learning topics cover Principal Component Analysis and clustering methods. We then introduce sampling and Markov Chain Monte Carlo, followed by Gaussian Processes as probabilistic nonparametric methods and neural networks in the broader statistical context. Our theory-focused approach derives each method from first principles with complete mathematical development and then shows its astronomical applications. We put statistical insight first and explain why algorithms work, when they are appropriate, and how they connect to broader statistical principles. From a solid grounding in classical methods and their theory, the treatment leads to modern techniques such as neural networks. With this grounding, readers can apply the methods to astronomical research with proper attention to assumptions, limits and uncertainty propagation, as large astronomical surveys require. The 22 companion tutorials apply the methods to astronomical data in Jupyter notebooks.
Tensor Network (TN) Kernel Machines speed up model learning by representing parameters as low-rank TNs, reducing computation and memory use. However, most TN-based Kernel methods are deterministic and ignore parameter uncertainty. Further, they require manual tuning of model complexity hyperparameters like tensor rank and feature dimensions, often through trial-and-error or computationally costly methods like cross-validation. We propose Bayesian Tensor Network Kernel Machines, a fully probabilistic framework that uses sparsity-inducing hierarchical priors on TN factors to automatically infer model complexity. This enables automatic inference of tensor rank and feature dimensions, while also identifying the most relevant features for prediction, thereby enhancing model interpretability. All the model parameters and hyperparameters are treated as latent variables with corresponding priors. Given the Bayesian approach and latent variable dependencies, we apply a mean-field variational inference to approximate their posteriors. We show that applying a mean-field approximation to TN factors yields a Bayesian ALS algorithm with the same computational complexity as its deterministic counterpart, enabling uncertainty quantification at no extra computational cost. Experiments on synthetic and real-world datasets demonstrate the superior performance of our model in prediction accuracy, uncertainty quantification, interpretability, and scalability.
Unlike classification, whose goal is to estimate the class of each data point in a dataset, prevalence estimation or quantification is a task that aims to estimate the distribution of classes in a dataset. The two main tasks in prevalence estimation are to adjust for bias, due to the prevalence in the training dataset, and to quantify the uncertainty in the estimate. The standard methods used to quantify uncertainty in prevalence estimates are bootstrapping and Bayesian quantification methods. It is not clear which approach is ideal in terms of precision (i.e. the width of confidence intervals) and coverage (i.e. the confidence intervals being well-calibrated). Here, we propose Precise Quantifier (PQ), a Bayesian quantifier that is more precise than existing quantifiers and with well-calibrated coverage. We discuss the theory behind PQ and present experiments based on simulated and real-world datasets. Through these experiments, we establish the factors which influence quantification precision: the discriminatory power of the underlying classifier; the size of the labeled dataset used to train the quantifier; and the size of the unlabeled dataset for which prevalence is estimated. Our analysis provides deep insights into uncertainty quantification for quantification learning.
Hallucinations, defined as instances where Large Language Models (LLMs) generate false or misleading content, pose a significant challenge that impacts the safety and trust of downstream applications. We introduce UQLM, a Python package for LLM hallucination detection using state-of-the-art uncertainty quantification (UQ) techniques. This toolkit offers a suite of UQ-based scorers that compute response-level confidence scores ranging from 0 to 1. This library provides an off-the-shelf solution for UQ-based hallucination detection that can be easily integrated to enhance the reliability of LLM outputs.
As Large Language Models (LLMs) are integrated into safety-critical applications involving sequential decision-making in the real world, it is essential to know when to trust LLM decisions. Existing LLM Uncertainty Quantification (UQ) methods are primarily designed for single-turn question-answering formats, resulting in multi-step decision-making scenarios, e.g., LLM agentic system, being underexplored. In this paper, we introduce a principled, information-theoretic framework that decomposes LLM sequential decision uncertainty into two parts: (i) internal uncertainty intrinsic to the current decision, which is focused on existing UQ methods, and (ii) extrinsic uncertainty, a Mutual-Information (MI) quantity describing how much uncertainty should be inherited from preceding decisions. We then propose UProp, an efficient and effective extrinsic uncertainty estimator that converts the direct estimation of MI to the estimation of Pointwise Mutual Information (PMI) over multiple Trajectory-Dependent Decision Processes (TDPs). UProp is evaluated over extensive multi-step decision-making benchmarks, e.g., AgentBench and HotpotQA, with state-of-the-art LLMs, e.g., GPT-4.1 and DeepSeek-V3. Experimental results demonstrate that UProp significantly outperforms existing single-turn UQ baselines equipped with thoughtful aggregation strategies. Moreover, we provide a comprehensive analysis of UProp, including sampling efficiency, potential applications, and intermediate uncertainty propagation, to demonstrate its effectiveness. Codes will be available at https://github.com/jinhaoduan/UProp.
Despite their widespread use, large language models (LLMs) are known to hallucinate incorrect information and be poorly calibrated. This makes the uncertainty quantification of these models of critical importance, especially in high-stakes domains, such as autonomy and healthcare. Prior work has made Bayesian deep learning-based approaches to this problem more tractable by performing inference over the low-rank adaptation (LoRA) parameters of a fine-tuned model. While effective, these approaches struggle to scale to larger LLMs due to requiring further additional parameters compared to LoRA. In this work we present $\textbf{Scala}$ble $\textbf{B}$ayesian $\textbf{L}$ow-Rank Adaptation via Stochastic Variational Subspace Inference (ScalaBL). We perform Bayesian inference in an $r$-dimensional subspace, for LoRA rank $r$. By repurposing the LoRA parameters as projection matrices, we are able to map samples from this subspace into the full weight space of the LLM. This allows us to learn all the parameters of our approach using stochastic variational inference. Despite the low dimensionality of our subspace, we are able to achieve competitive performance with state-of-the-art approaches while only requiring ${\sim}1000$ additional parameters. Furthermore, it allows us to scale up to the largest Bayesian LLM to date, with four times as a many base parameters as prior work.
Deep object pose estimators are notoriously overconfident. A grasping agent that both estimates the 6-DoF pose of a target object and predicts the uncertainty of its own estimate could avoid task failure by choosing not to act under high uncertainty. Even though object pose estimation improves and uncertainty quantification research continues to make strides, few studies have connected them to the downstream task of robotic grasping. We propose a method for training lightweight, deep networks to predict whether a grasp guided by an image-based pose estimate will succeed before that grasp is attempted. We generate training data for our networks via object pose estimation on real images and simulated grasping. We also find that, despite high object variability in grasping trials, networks benefit from training on all objects jointly, suggesting that a diverse variety of objects can nevertheless contribute to the same goal.
Graph Neural Networks (GNNs) have shown remarkable performance in the healthcare domain. However, what remained challenging is quantifying the predictive uncertainty of GNNs, which is an important aspect of trustworthiness in clinical settings. While Bayesian and ensemble methods can be used to quantify uncertainty, they are computationally expensive. Additionally, the disagreement metric used by ensemble methods to compute uncertainty cannot capture the diversity of models in an ensemble network. In this paper, we propose a novel method, based on knowledge distillation, to quantify GNNs' uncertainty more efficiently and with higher precision. We apply self-distillation, where the same network serves as both the teacher and student models, thereby avoiding the need to train several networks independently. To ensure the impact of self-distillation, we develop an uncertainty metric that captures the diverse nature of the network by assigning different weights to each GNN classifier. We experimentally evaluate the precision, performance, and ability of our approach in distinguishing out-of-distribution data on two graph datasets: MIMIC-IV and Enzymes. The evaluation results demonstrate that the proposed method can effectively capture the predictive uncertainty of the model while having performance similar to that of the MC Dropout and ensemble methods. The code is publicly available at https://github.com/tailabTMU/UQ_GNN.
Deep neural networks often learn and rely on spurious correlations, i.e., superficial associations between non-causal features and the targets. For instance, an image classifier may identify camels based on the desert backgrounds. While it can yield high overall accuracy during training, it degrades generalization on more diverse scenarios where such correlations do not hold. This problem poses significant challenges for out-of-distribution robustness and trustworthiness. Existing methods typically mitigate this issue by using external group annotations or auxiliary deterministic models to learn unbiased representations. However, such information is costly to obtain, and deterministic models may fail to capture the full spectrum of biases learned by the models. To address these limitations, we propose Evidential Alignment, a novel framework that leverages uncertainty quantification to understand the behavior of the biased models without requiring group annotations. By quantifying the evidence of model prediction with second-order risk minimization and calibrating the biased models with the proposed evidential calibration technique, Evidential Alignment identifies and suppresses spurious correlations while preserving core features. We theoretically justify the effectiveness of our method as capable of learning the patterns of biased models and debiasing the model without requiring any spurious correlation annotations. Empirical results demonstrate that our method significantly improves group robustness across diverse architectures and data modalities, providing a scalable and principled solution to spurious correlations.
Vine copulas are sophisticated models for multivariate distributions and are increasingly used in machine learning. To facilitate their integration into modern ML pipelines, we introduce the vine computational graph, a DAG that abstracts the multilevel vine structure and associated computations. On this foundation, we devise new algorithms for conditional sampling, efficient sampling-order scheduling, and constructing vine structures for customized conditioning variables. We implement these ideas in torchvinecopulib, a GPU-accelerated Python library built upon PyTorch, delivering improved scalability for fitting, sampling, and density evaluation. Our experiments illustrate how gradient flowing through the vine can improve Vine Copula Autoencoders and that incorporating vines for uncertainty quantification in deep learning can outperform MC-dropout, deep ensembles, and Bayesian Neural Networks in sharpness, calibration, and runtime. By recasting vine copula models as computational graphs, our work connects classical dependence modeling with modern deep-learning toolchains and facilitates the integration of state-of-the-art copula methods in modern machine learning pipelines.
State estimation in stochastic dynamical systems with noisy measurements is a challenge. While the Kalman filter is optimal for linear systems with independent Gaussian white noise, real-world conditions often deviate from these assumptions, prompting the rise of data-driven filtering techniques. This paper introduces Recursive KalmanNet, a Kalman-filter-informed recurrent neural network designed for accurate state estimation with consistent error covariance quantification. Our approach propagates error covariance using the recursive Joseph's formula and optimizes the Gaussian negative log-likelihood. Experiments with non-Gaussian measurement white noise demonstrate that our model outperforms both the conventional Kalman filter and an existing state-of-the-art deep learning based estimator.
We introduce an approach to topic modelling with document-level covariates that remains tractable in the face of large text corpora. This is achieved by de-emphasizing the role of parameter estimation in an underlying probabilistic model, assuming instead that the data come from a fixed but unknown distribution whose statistical functionals are of interest. We propose combining a convex formulation of non-negative matrix factorization with standard regression techniques as a fast-to-compute and useful estimate of such a functional. Uncertainty quantification can then be achieved by reposing non-parametric resampling methods on top of this scheme. This is in contrast to popular topic modelling paradigms, which posit a complex and often hard-to-fit generative model of the data. We argue that the simple, non-parametric approach advocated here is faster, more interpretable, and enjoys better inferential justification than said generative models. Finally, our methods are demonstrated with an application analysing covariate effects on discourse of flavours attributed to Canadian beers.
Population synthesis simulations of compact binary coalescences~(CBCs) play a crucial role in extracting astrophysical insights from an ensemble of gravitational wave~(GW) observations. However, realistic simulations are costly to implement for a dense grid of initial conditions. Normalizing flows can emulate the distribution functions of a simulated population of binary parameters and thereby enable empirical constraints on the astrophysical initial conditions and branching fractions of various formation channels given data from a catalog of GW observations. They can also be used for data amplification in sparse regions of the CBC parameter space to guide the development of phenomenological population models for rarely synthesizable systems with components in theorized mass gaps, without having to simulate a prohibitively large number of binaries. But flow predictions are wrought with uncertainties, especially for sparse training sets. In this work I develop a method for quantifying and marginalizing uncertainties in the emulators by introducing the Bayesian Normalizing flow, a conditional density estimator constructed from Bayesian neural networks. Using the exact likelihood function associated with density estimators I sample the posterior distribution of flow parameters with suitably chosen priors to quantify and marginalize over flow uncertainties. I demonstrate the accuracy, calibration, and data-amplification impacts of the estimated uncertainties for simulations of binary black hole populations formed through common envelope evolution. I outline applications of the methodology in simulation-based inference from growing GW catalogs and sketch other uses for general simulation-based approaches in GW astronomy.
Large language models (LLMs) have transformed natural language processing, but their reliable deployment requires effective uncertainty quantification (UQ). Existing UQ methods are often heuristic and lack a probabilistic foundation. This paper begins by providing a theoretical justification for the role of perturbations in UQ for LLMs. We then introduce a dual random walk perspective, modeling input-output pairs as two Markov chains with transition probabilities defined by semantic similarity. Building on this, we propose a fully probabilistic framework based on an inverse model, which quantifies uncertainty by evaluating the diversity of the input space conditioned on a given output through systematic perturbations. Within this framework, we define a new uncertainty measure, Inv-Entropy. A key strength of our framework is its flexibility: it supports various definitions of uncertainty measures, embeddings, perturbation strategies, and similarity metrics. We also propose GAAP, a perturbation algorithm based on genetic algorithms, which enhances the diversity of sampled inputs. In addition, we introduce a new evaluation metric, Temperature Sensitivity of Uncertainty (TSU), which directly assesses uncertainty without relying on correctness as a proxy. Extensive experiments demonstrate that Inv-Entropy outperforms existing semantic UQ methods. The code to reproduce the results can be found at https://github.com/UMDataScienceLab/Uncertainty-Quantification-for-LLMs.
Accurate probabilistic weather forecasting demands both high accuracy and efficient uncertainty quantification, challenges that overburden both ensemble numerical weather prediction (NWP) and recent machine-learning methods. We introduce LaDCast, the first global latent-diffusion framework for medium-range ensemble forecasting, which generates hourly ensemble forecasts entirely in a learned latent space. An autoencoder compresses high-dimensional ERA5 reanalysis fields into a compact representation, and a transformer-based diffusion model produces sequential latent updates with arbitrary hour initialization. The model incorporates Geometric Rotary Position Embedding (GeoRoPE) to account for the Earth's spherical geometry, a dual-stream attention mechanism for efficient conditioning, and sinusoidal temporal embeddings to capture seasonal patterns. LaDCast achieves deterministic and probabilistic skill close to that of the European Centre for Medium-Range Forecast IFS-ENS, without any explicit perturbations. Notably, LaDCast demonstrates superior performance in tracking rare extreme events such as cyclones, capturing their trajectories more accurately than established models. By operating in latent space, LaDCast reduces storage and compute by orders of magnitude, demonstrating a practical path toward forecasting at kilometer-scale resolution in real time. We open-source our code and models and provide the training and evaluation pipelines at: https://github.com/tonyzyl/ladcast.
Word-level quality estimation (WQE) aims to automatically identify fine-grained error spans in machine-translated outputs and has found many uses, including assisting translators during post-editing. Modern WQE techniques are often expensive, involving prompting of large language models or ad-hoc training on large amounts of human-labeled data. In this work, we investigate efficient alternatives exploiting recent advances in language model interpretability and uncertainty quantification to identify translation errors from the inner workings of translation models. In our evaluation spanning 14 metrics across 12 translation directions, we quantify the impact of human label variation on metric performance by using multiple sets of human labels. Our results highlight the untapped potential of unsupervised metrics, the shortcomings of supervised methods when faced with label uncertainty, and the brittleness of single-annotator evaluation practices.
Conformal prediction (CP) is a popular frequentist framework for representing uncertainty by providing prediction sets that guarantee coverage of the true label with a user-adjustable probability. In most applications, CP operates on confidence scores coming from a standard (first-order) probabilistic predictor (e.g., softmax outputs). Second-order predictors, such as credal set predictors or Bayesian models, are also widely used for uncertainty quantification and are known for their ability to represent both aleatoric and epistemic uncertainty. Despite their popularity, there is still an open question on ``how they can be incorporated into CP''. In this paper, we discuss the desiderata for CP when valid second-order predictions are available. We then introduce Bernoulli prediction sets (BPS), which produce the smallest prediction sets that ensure conditional coverage in this setting. When given first-order predictions, BPS reduces to the well-known adaptive prediction sets (APS). Furthermore, when the validity assumption on the second-order predictions is compromised, we apply conformal risk control to obtain a marginal coverage guarantee while still accounting for epistemic uncertainty.
Conformal Prediction (CP) is a popular method for uncertainty quantification with machine learning models. While conformal prediction provides probabilistic guarantees regarding the coverage of the true label, these guarantees are agnostic to the presence of sensitive attributes within the dataset. In this work, we formalize \textit{Conformal Fairness}, a notion of fairness using conformal predictors, and provide a theoretically well-founded algorithm and associated framework to control for the gaps in coverage between different sensitive groups. Our framework leverages the exchangeability assumption (implicit to CP) rather than the typical IID assumption, allowing us to apply the notion of Conformal Fairness to data types and tasks that are not IID, such as graph data. Experiments were conducted on graph and tabular datasets to demonstrate that the algorithm can control fairness-related gaps in addition to coverage aligned with theoretical expectations.
Survival analysis, which estimates the probability of event occurrence over time from censored data, is fundamental in numerous real-world applications, particularly in high-stakes domains such as healthcare and risk assessment. Despite advances in numerous survival models, quantifying the uncertainty of predictions from these models remains underexplored and challenging. The lack of reliable uncertainty quantification limits the interpretability and trustworthiness of survival models, hindering their adoption in clinical decision-making and other sensitive applications. To bridge this gap, in this work, we introduce SurvUnc, a novel meta-model based framework for post-hoc uncertainty quantification for survival models. SurvUnc introduces an anchor-based learning strategy that integrates concordance knowledge into meta-model optimization, leveraging pairwise ranking performance to estimate uncertainty effectively. Notably, our framework is model-agnostic, ensuring compatibility with any survival model without requiring modifications to its architecture or access to its internal parameters. Especially, we design a comprehensive evaluation pipeline tailored to this critical yet overlooked problem. Through extensive experiments on four publicly available benchmarking datasets and five representative survival models, we demonstrate the superiority of SurvUnc across multiple evaluation scenarios, including selective prediction, misprediction detection, and out-of-domain detection. Our results highlight the effectiveness of SurvUnc in enhancing model interpretability and reliability, paving the way for more trustworthy survival predictions in real-world applications.
This paper explores Uncertainty Quantification (UQ) in SVM predictions, particularly for regression and forecasting tasks. Unlike the Neural Network, the SVM solutions are typically more stable, sparse, optimal and interpretable. However, there are only few literature which addresses the UQ in SVM prediction. At first, we provide a comprehensive summary of existing Prediction Interval (PI) estimation and probabilistic forecasting methods developed in the SVM framework and evaluate them against the key properties expected from an ideal PI model. We find that none of the existing SVM PI models achieves a sparse solution. To introduce sparsity in SVM model, we propose the Sparse Support Vector Quantile Regression (SSVQR) model, which constructs PIs and probabilistic forecasts by solving a pair of linear programs. Further, we develop a feature selection algorithm for PI estimation using SSVQR that effectively eliminates a significant number of features while improving PI quality in case of high-dimensional dataset. Finally we extend the SVM models in Conformal Regression setting for obtaining more stable prediction set with finite test set guarantees. Extensive experiments on artificial, real-world benchmark datasets compare the different characteristics of both existing and proposed SVM-based PI estimation methods and also highlight the advantages of the feature selection in PI estimation. Furthermore, we compare both, the existing and proposed SVM-based PI estimation models, with modern deep learning models for probabilistic forecasting tasks on benchmark datasets. Furthermore, SVM models show comparable or superior performance to modern complex deep learning models for probabilistic forecasting task in our experiments.