Daily Papers

Streaming anomaly detection requires algorithms that operate under strict constraints: bounded memory, single-pass processing, and constant-time complexity. We present PySAD, a comprehensive Python framework addressing these challenges through a unified architecture. The framework implements 17+ streaming algorithms (LODA, Half-Space Trees, xStream) with specialized components including projectors, probability calibrators, and postprocessors. Unlike existing batch-focused frameworks, PySAD enables efficient real-time processing with bounded memory while maintaining compatibility with PyOD and scikit-learn. Supporting all learning paradigms for univariate and multivariate streams, PySAD provides the most comprehensive streaming anomaly detection toolkit in Python. The source code is publicly available at github.com/selimfirat/pysad.

Multi-class classification methods that produce sets of probabilistic classifiers, such as ensemble learning methods, are able to model aleatoric and epistemic uncertainty. Aleatoric uncertainty is then typically quantified via the Bayes error, and epistemic uncertainty via the size of the set. In this paper, we extend the notion of calibration, which is commonly used to evaluate the validity of the aleatoric uncertainty representation of a single probabilistic classifier, to assess the validity of an epistemic uncertainty representation obtained by sets of probabilistic classifiers. Broadly speaking, we call a set of probabilistic classifiers calibrated if one can find a calibrated convex combination of these classifiers. To evaluate this notion of calibration, we propose a novel nonparametric calibration test that generalizes an existing test for single probabilistic classifiers to the case of sets of probabilistic classifiers. Making use of this test, we empirically show that ensembles of deep neural networks are often not well calibrated.

Calibration measures and reliability diagrams are two fundamental tools for measuring and interpreting the calibration of probabilistic predictors. Calibration measures quantify the degree of miscalibration, and reliability diagrams visualize the structure of this miscalibration. However, the most common constructions of reliability diagrams and calibration measures -- binning and ECE -- both suffer from well-known flaws (e.g. discontinuity). We show that a simple modification fixes both constructions: first smooth the observations using an RBF kernel, then compute the Expected Calibration Error (ECE) of this smoothed function. We prove that with a careful choice of bandwidth, this method yields a calibration measure that is well-behaved in the sense of (B{\l}asiok, Gopalan, Hu, and Nakkiran 2023a) -- a consistent calibration measure. We call this measure the SmoothECE. Moreover, the reliability diagram obtained from this smoothed function visually encodes the SmoothECE, just as binned reliability diagrams encode the BinnedECE. We also provide a Python package with simple, hyperparameter-free methods for measuring and plotting calibration: `pip install relplot\`.

Confidence calibration -- the problem of predicting probability estimates representative of the true correctness likelihood -- is important for classification models in many applications. We discover that modern neural networks, unlike those from a decade ago, are poorly calibrated. Through extensive experiments, we observe that depth, width, weight decay, and Batch Normalization are important factors influencing calibration. We evaluate the performance of various post-processing calibration methods on state-of-the-art architectures with image and document classification datasets. Our analysis and experiments not only offer insights into neural network learning, but also provide a simple and straightforward recipe for practical settings: on most datasets, temperature scaling -- a single-parameter variant of Platt Scaling -- is surprisingly effective at calibrating predictions.

Deep neural networks have demonstrated remarkable performance across numerous learning tasks but often suffer from miscalibration, resulting in unreliable probability outputs. This has inspired many recent works on mitigating miscalibration, particularly through post-hoc recalibration methods that aim to obtain calibrated probabilities without sacrificing the classification performance of pre-trained models. In this study, we summarize and categorize previous works into three general strategies: intuitively designed methods, binning-based methods, and methods based on formulations of ideal calibration. Through theoretical and practical analysis, we highlight ten common limitations in previous approaches. To address these limitations, we propose a probabilistic learning framework for calibration called h-calibration, which theoretically constructs an equivalent learning formulation for canonical calibration with boundedness. On this basis, we design a simple yet effective post-hoc calibration algorithm. Our method not only overcomes the ten identified limitations but also achieves markedly better performance than traditional methods, as validated by extensive experiments. We further analyze, both theoretically and experimentally, the relationship and advantages of our learning objective compared to traditional proper scoring rule. In summary, our probabilistic framework derives an approximately equivalent differentiable objective for learning error-bounded calibrated probabilities, elucidating the correspondence and convergence properties of computational statistics with respect to theoretical bounds in canonical calibration. The theoretical effectiveness is verified on standard post-hoc calibration benchmarks by achieving state-of-the-art performance. This research offers valuable reference for learning reliable likelihood in related fields.

Accurate probabilistic predictions are essential for optimal decision making. While neural network miscalibration has been studied primarily in classification, we investigate this in the less-explored domain of regression. We conduct the largest empirical study to date to assess the probabilistic calibration of neural networks. We also analyze the performance of recalibration, conformal, and regularization methods to enhance probabilistic calibration. Additionally, we introduce novel differentiable recalibration and regularization methods, uncovering new insights into their effectiveness. Our findings reveal that regularization methods offer a favorable tradeoff between calibration and sharpness. Post-hoc methods exhibit superior probabilistic calibration, which we attribute to the finite-sample coverage guarantee of conformal prediction. Furthermore, we demonstrate that quantile recalibration can be considered as a specific case of conformal prediction. Our study is fully reproducible and implemented in a common code base for fair comparisons.

Recent advances in machine learning have significantly improved prediction accuracy in various applications. However, ensuring the calibration of probabilistic predictions remains a significant challenge. Despite efforts to enhance model calibration, the rigorous statistical evaluation of model calibration remains less explored. In this work, we develop confidence intervals the ell_2 Expected Calibration Error (ECE). We consider top-1-to-k calibration, which includes both the popular notion of confidence calibration as well as full calibration. For a debiased estimator of the ECE, we show asymptotic normality, but with different convergence rates and asymptotic variances for calibrated and miscalibrated models. We develop methods to construct asymptotically valid confidence intervals for the ECE, accounting for this behavior as well as non-negativity. Our theoretical findings are supported through extensive experiments, showing that our methods produce valid confidence intervals with shorter lengths compared to those obtained by resampling-based methods.

Quantifying the uncertainty of predictions made by large language models (LLMs) in binary text classification tasks remains a challenge. Calibration, in the context of LLMs, refers to the alignment between the model's predicted probabilities and the actual correctness of its predictions. A well-calibrated model should produce probabilities that accurately reflect the likelihood of its predictions being correct. We propose a novel approach that utilizes the inductive Venn--Abers predictor (IVAP) to calibrate the probabilities associated with the output tokens corresponding to the binary labels. Our experiments on the BoolQ dataset using the Llama 2 model demonstrate that IVAP consistently outperforms the commonly used temperature scaling method for various label token choices, achieving well-calibrated probabilities while maintaining high predictive quality. Our findings contribute to the understanding of calibration techniques for LLMs and provide a practical solution for obtaining reliable uncertainty estimates in binary question answering tasks, enhancing the interpretability and trustworthiness of LLM predictions.

As LLM-based judges become integral to industry applications, obtaining well-calibrated uncertainty estimates efficiently has become critical for production deployment. However, existing techniques, such as verbalized confidence and multi-generation methods, are often either poorly calibrated or computationally expensive. We introduce linear probes trained with a Brier score-based loss to provide calibrated uncertainty estimates from reasoning judges' hidden states, requiring no additional model training. We evaluate our approach on both objective tasks (reasoning, mathematics, factuality, coding) and subjective human preference judgments. Our results demonstrate that probes achieve superior calibration compared to existing methods with approx10x computational savings, generalize robustly to unseen evaluation domains, and deliver higher accuracy on high-confidence predictions. However, probes produce conservative estimates that underperform on easier datasets but may benefit safety-critical deployments prioritizing low false-positive rates. Overall, our work demonstrates that interpretability-based uncertainty estimation provides a practical and scalable plug-and-play solution for LLM judges in production.

Reliable probability estimation is of crucial importance in many real-world applications where there is inherent (aleatoric) uncertainty. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the difference that the objective is to estimate probabilities rather than predicting the specific outcome. This work investigates probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on model (epistemic) uncertainty. For problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty: precipitation forecasting from radar images, predicting cancer patient survival from histopathology images, and predicting car crashes from dashcam videos. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.

Reliable usage of object detectors require them to be calibrated -- a crucial problem that requires careful attention. Recent approaches towards this involve (1) designing new loss functions to obtain calibrated detectors by training them from scratch, and (2) post-hoc Temperature Scaling (TS) that learns to scale the likelihood of a trained detector to output calibrated predictions. These approaches are then evaluated based on a combination of Detection Expected Calibration Error (D-ECE) and Average Precision. In this work, via extensive analysis and insights, we highlight that these recent evaluation frameworks, evaluation metrics, and the use of TS have notable drawbacks leading to incorrect conclusions. As a step towards fixing these issues, we propose a principled evaluation framework to jointly measure calibration and accuracy of object detectors. We also tailor efficient and easy-to-use post-hoc calibration approaches such as Platt Scaling and Isotonic Regression specifically for object detection task. Contrary to the common notion, our experiments show that once designed and evaluated properly, post-hoc calibrators, which are extremely cheap to build and use, are much more powerful and effective than the recent train-time calibration methods. To illustrate, D-DETR with our post-hoc Isotonic Regression calibrator outperforms the recent train-time state-of-the-art calibration method Cal-DETR by more than 7 D-ECE on the COCO dataset. Additionally, we propose improved versions of the recently proposed Localization-aware ECE and show the efficacy of our method on these metrics as well. Code is available at: https://github.com/fiveai/detection_calibration.

Although Neural Radiance Fields (NeRFs) have markedly improved novel view synthesis, accurate uncertainty quantification in their image predictions remains an open problem. The prevailing methods for estimating uncertainty, including the state-of-the-art Density-aware NeRF Ensembles (DANE) [29], quantify uncertainty without calibration. This frequently leads to over- or under-confidence in image predictions, which can undermine their real-world applications. In this paper, we propose a method which, for the first time, achieves calibrated uncertainties for NeRFs. To accomplish this, we overcome a significant challenge in adapting existing calibration techniques to NeRFs: a need to hold out ground truth images from the target scene, reducing the number of images left to train the NeRF. This issue is particularly problematic in sparse-view settings, where we can operate with as few as three images. To address this, we introduce the concept of a meta-calibrator that performs uncertainty calibration for NeRFs with a single forward pass without the need for holding out any images from the target scene. Our meta-calibrator is a neural network that takes as input the NeRF images and uncalibrated uncertainty maps and outputs a scene-specific calibration curve that corrects the NeRF's uncalibrated uncertainties. We show that the meta-calibrator can generalize on unseen scenes and achieves well-calibrated and state-of-the-art uncertainty for NeRFs, significantly beating DANE and other approaches. This opens opportunities to improve applications that rely on accurate NeRF uncertainty estimates such as next-best view planning and potentially more trustworthy image reconstruction for medical diagnosis. The code is available at https://niki-amini-naieni.github.io/instantcalibration.github.io/.

Model calibration aims to align confidence with prediction correctness. The Cross-Entropy (CE) loss is widely used for calibrator training, which enforces the model to increase confidence on the ground truth class. However, we find the CE loss has intrinsic limitations. For example, for a narrow misclassification, a calibrator trained by the CE loss often produces high confidence on the wrongly predicted class (e.g., a test sample is wrongly classified and its softmax score on the ground truth class is around 0.4), which is undesirable. In this paper, we propose a new post-hoc calibration objective derived from the aim of calibration. Intuitively, the proposed objective function asks that the calibrator decrease model confidence on wrongly predicted samples and increase confidence on correctly predicted samples. Because a sample itself has insufficient ability to indicate correctness, we use its transformed versions (e.g., rotated, greyscaled and color-jittered) during calibrator training. Trained on an in-distribution validation set and tested with isolated, individual test samples, our method achieves competitive calibration performance on both in-distribution and out-of-distribution test sets compared with the state of the art. Further, our analysis points out the difference between our method and commonly used objectives such as CE loss and mean square error loss, where the latters sometimes deviates from the calibration aim.

Probability estimation models play an important role in various fields, such as weather forecasting, recommendation systems, and sports analysis. Among several models estimating probabilities, it is difficult to evaluate which model gives reliable probabilities since the ground-truth probabilities are not available. The win probability estimation model for esports, which calculates the win probability under a certain game state, is also one of the fields being actively studied in probability estimation. However, most of the previous works evaluated their models using accuracy, a metric that only can measure the performance of discrimination. In this work, we firstly investigate the Brier score and the Expected Calibration Error (ECE) as a replacement of accuracy used as a performance evaluation metric for win probability estimation models in esports field. Based on the analysis, we propose a novel metric called Balance score which is a simple yet effective metric in terms of six good properties that probability estimation metric should have. Under the general condition, we also found that the Balance score can be an effective approximation of the true expected calibration error which has been imperfectly approximated by ECE using the binning technique. Extensive evaluations using simulation studies and real game snapshot data demonstrate the promising potential to adopt the proposed metric not only for the win probability estimation model for esports but also for evaluating general probability estimation models.

A probabilistic classifier with reliable predictive uncertainties i) fits successfully to the target domain data, ii) provides calibrated class probabilities in difficult regions of the target domain (e.g.\ class overlap), and iii) accurately identifies queries coming out of the target domain and rejects them. We introduce an original combination of Evidential Deep Learning, Neural Processes, and Neural Turing Machines capable of providing all three essential properties mentioned above for total uncertainty quantification. We observe our method on five classification tasks to be the only one that can excel all three aspects of total calibration with a single standalone predictor. Our unified solution delivers an implementation-friendly and compute efficient recipe for safety clearance and provides intellectual economy to an investigation of algorithmic roots of epistemic awareness in deep neural nets.

Recently, diffusion probabilistic models (DPMs) have achieved promising results in diverse generative tasks. A typical DPM framework includes a forward process that gradually diffuses the data distribution and a reverse process that recovers the data distribution from time-dependent data scores. In this work, we observe that the stochastic reverse process of data scores is a martingale, from which concentration bounds and the optional stopping theorem for data scores can be derived. Then, we discover a simple way for calibrating an arbitrary pretrained DPM, with which the score matching loss can be reduced and the lower bounds of model likelihood can consequently be increased. We provide general calibration guidelines under various model parametrizations. Our calibration method is performed only once and the resulting models can be used repeatedly for sampling. We conduct experiments on multiple datasets to empirically validate our proposal. Our code is at https://github.com/thudzj/Calibrated-DPMs.

We show that a GPT-3 model can learn to express uncertainty about its own answers in natural language -- without use of model logits. When given a question, the model generates both an answer and a level of confidence (e.g. "90% confidence" or "high confidence"). These levels map to probabilities that are well calibrated. The model also remains moderately calibrated under distribution shift, and is sensitive to uncertainty in its own answers, rather than imitating human examples. To our knowledge, this is the first time a model has been shown to express calibrated uncertainty about its own answers in natural language. For testing calibration, we introduce the CalibratedMath suite of tasks. We compare the calibration of uncertainty expressed in words ("verbalized probability") to uncertainty extracted from model logits. Both kinds of uncertainty are capable of generalizing calibration under distribution shift. We also provide evidence that GPT-3's ability to generalize calibration depends on pre-trained latent representations that correlate with epistemic uncertainty over its answers.

A trustworthy real-world prediction system should produce well-calibrated confidence scores; that is, its confidence in an answer should be indicative of the likelihood that the answer is correct, enabling deferral to an expert in cases of low-confidence predictions. Recent studies have shown that unsupervised pre-training produces large language models (LMs) whose conditional probabilities are remarkably well-calibrated. However, the most widely-used LMs are fine-tuned with reinforcement learning from human feedback (RLHF-LMs), and some studies have suggested that RLHF-LMs produce conditional probabilities that are very poorly calibrated. In light of this perceived weakness, we conduct a broad evaluation of methods for extracting confidence scores from RLHF-LMs. For RLHF-LMs such as ChatGPT, GPT-4, and Claude, we find that verbalized confidences emitted as output tokens are typically better-calibrated than the model's conditional probabilities on the TriviaQA, SciQ, and TruthfulQA benchmarks, often reducing the expected calibration error by a relative 50%.

Accurate sensor calibration is crucial for autonomous systems, yet its uncertainty quantification remains underexplored. We present the first approach to integrate uncertainty awareness into online extrinsic calibration, combining Monte Carlo Dropout with Conformal Prediction to generate prediction intervals with a guaranteed level of coverage. Our method proposes a framework to enhance existing calibration models with uncertainty quantification, compatible with various network architectures. Validated on KITTI (RGB Camera-LiDAR) and DSEC (Event Camera-LiDAR) datasets, we demonstrate effectiveness across different visual sensor types, measuring performance with adapted metrics to evaluate the efficiency and reliability of the intervals. By providing calibration parameters with quantifiable confidence measures, we offer insights into the reliability of calibration estimates, which can greatly improve the robustness of sensor fusion in dynamic environments and usefully serve the Computer Vision community.

Process reward models (PRMs) play a central role in guiding inference-time scaling algorithms for large language models (LLMs). However, we observe that even state-of-the-art PRMs can be poorly calibrated and often overestimate success probabilities. To address this, we present a calibration approach, performed via quantile regression, that adjusts PRM outputs to better align with true success probabilities. Leveraging these calibrated success estimates and their associated confidence bounds, we introduce an instance-adaptive scaling (IAS) framework that dynamically adjusts the inference budget based on the estimated likelihood that a partial reasoning trajectory will yield a correct final answer. Unlike conventional methods that allocate a fixed number of reasoning trajectories per query, this approach successfully adapts to each instance and reasoning step when using our calibrated PRMs. Experiments on mathematical reasoning benchmarks show that (i) our PRM calibration method successfully achieves small calibration error, outperforming the baseline methods, (ii) calibration is crucial for enabling effective adaptive scaling, and (iii) the proposed IAS strategy reduces inference costs while maintaining final answer accuracy, utilizing less compute on more confident problems as desired.

Identifying how much a model {p}_{theta}(Y|X) knows about the stochastic real-world process p(Y|X) it was trained on is important to ensure it avoids producing incorrect or "hallucinated" answers or taking unsafe actions. But this is difficult for generative models because probabilistic predictions do not distinguish between per-response noise (aleatoric uncertainty) and lack of knowledge about the process (epistemic uncertainty), and existing epistemic uncertainty quantification techniques tend to be overconfident when the model underfits. We propose a general strategy for teaching a model to both approximate p(Y|X) and also estimate the remaining gaps between {p}_{theta}(Y|X) and p(Y|X): train it to predict pairs of independent responses drawn from the true conditional distribution, allow it to "cheat" by observing one response while predicting the other, then measure how much it cheats. Remarkably, we prove that being good at cheating (i.e. cheating whenever it improves your prediction) is equivalent to being second-order calibrated, a principled extension of ordinary calibration that allows us to construct provably-correct frequentist confidence intervals for p(Y|X) and detect incorrect responses with high probability. We demonstrate empirically that our approach accurately estimates how much models don't know across ambiguous image classification, (synthetic) language modeling, and partially-observable navigation tasks, outperforming existing techniques.

Predicting not only the target but also an accurate measure of uncertainty is important for many machine learning applications and in particular safety-critical ones. In this work we study the calibration of uncertainty prediction for regression tasks which often arise in real-world systems. We show that the existing definition for calibration of a regression uncertainty [Kuleshov et al. 2018] has severe limitations in distinguishing informative from non-informative uncertainty predictions. We propose a new definition that escapes this caveat and an evaluation method using a simple histogram-based approach. Our method clusters examples with similar uncertainty prediction and compares the prediction with the empirical uncertainty on these examples. We also propose a simple, scaling-based calibration method that preforms as well as much more complex ones. We show results on both a synthetic, controlled problem and on the object detection bounding-box regression task using the COCO and KITTI datasets.

Despite their impressive predictive performance in various computer vision tasks, deep neural networks (DNNs) tend to make overly confident predictions, which hinders their widespread use in safety-critical applications. While there have been recent attempts to calibrate DNNs, most of these efforts have primarily been focused on classification tasks, thus neglecting DNN-based object detectors. Although several recent works addressed calibration for object detection and proposed differentiable penalties, none of them are consistent estimators of established concepts in calibration. In this work, we tackle the challenge of defining and estimating calibration error specifically for this task. In particular, we adapt the definition of classification calibration error to handle the nuances associated with object detection, and predictions in structured output spaces more generally. Furthermore, we propose a consistent and differentiable estimator of the detection calibration error, utilizing kernel density estimation. Our experiments demonstrate the effectiveness of our estimator against competing train-time and post-hoc calibration methods, while maintaining similar detection performance.

Large Language Models (LLMs) are increasingly deployed in high-stakes domains such as science, law, and healthcare, where accurate expressions of uncertainty are essential for reliability and trust. However, current LLMs are often observed to generate incorrect answers with high confidence, a phenomenon known as "overconfidence". Recent efforts have focused on calibrating LLMs' verbalized confidence: i.e., their expressions of confidence in text form, such as "I am 80% confident that...". Existing approaches either rely on prompt engineering or fine-tuning with heuristically generated uncertainty estimates, both of which have limited effectiveness and generalizability. Motivated by the notion of proper scoring rules for calibration in classical machine learning models, we introduce ConfTuner, a simple and efficient fine-tuning method that introduces minimal overhead and does not require ground-truth confidence scores or proxy confidence estimates. ConfTuner relies on a new loss function, tokenized Brier score, which we theoretically prove to be a proper scoring rule, intuitively meaning that it "correctly incentivizes the model to report its true probability of being correct". ConfTuner improves calibration across diverse reasoning tasks and generalizes to black-box models such as GPT-4o. Our results further show that better-calibrated confidence enables downstream gains in self-correction and model cascade, advancing the development of trustworthy LLM systems. The code is available at https://github.com/liushiliushi/ConfTuner.

Transformers, currently the state-of-the-art in natural language understanding (NLU) tasks, are prone to generate uncalibrated predictions or extreme probabilities, making the process of taking different decisions based on their output relatively difficult. In this paper we propose to build several inductive Venn--ABERS predictors (IVAP), which are guaranteed to be well calibrated under minimal assumptions, based on a selection of pre-trained transformers. We test their performance over a set of diverse NLU tasks and show that they are capable of producing well-calibrated probabilistic predictions that are uniformly spread over the [0,1] interval -- all while retaining the original model's predictive accuracy.

Calibrated uncertainty estimates in machine learning are crucial to many fields such as autonomous vehicles, medicine, and weather and climate forecasting. While there is extensive literature on uncertainty calibration for classification, the classification findings do not always translate to regression. As a result, modern models for predicting uncertainty in regression settings typically produce uncalibrated and overconfident estimates. To address these gaps, we present a calibration method for regression settings that does not assume a particular uncertainty distribution over the error: Calibrating Regression Uncertainty Distributions Empirically (CRUDE). CRUDE makes the weaker assumption that error distributions have a constant arbitrary shape across the output space, shifted by predicted mean and scaled by predicted standard deviation. We detail a theoretical connection between CRUDE and conformal inference. Across an extensive set of regression tasks, CRUDE demonstrates consistently sharper, better calibrated, and more accurate uncertainty estimates than state-of-the-art techniques.

Uncertainty quantification (UQ) is vital for trustworthy deep learning, yet existing methods are either computationally intensive, such as Bayesian or ensemble methods, or provide only partial, task-specific estimates, such as single-forward-pass techniques. In this paper, we propose a post-hoc single-forward-pass framework that jointly captures aleatoric and epistemic uncertainty without modifying or retraining pretrained models. Our method applies Split-Point Analysis (SPA) to decompose predictive residuals into upper and lower subsets, computing Mean Absolute Residuals (MARs) on each side. We prove that, under ideal conditions, the total MAR equals the harmonic mean of subset MARs; deviations define a novel Self-consistency Discrepancy Score (SDS) for fine-grained epistemic estimation across regression and classification. For regression, side-specific quantile regression yields prediction intervals with improved empirical coverage, which are further calibrated via SDS. For classification, when calibration data are available, we apply SPA-based calibration identities to adjust the softmax outputs and then compute predictive entropy on these calibrated probabilities. Extensive experiments on diverse regression and classification benchmarks demonstrate that our framework matches or exceeds several state-of-the-art UQ methods while incurring minimal overhead. Our source code is available at https://github.com/zzz0527/SPC-UQ.

LLM-as-judge evaluation has become the de facto standard for scaling model assessment, but the practice is statistically unsound: uncalibrated scores can invert preferences, naive confidence intervals on uncalibrated scores achieve near-0% coverage, and importance-weighted estimators collapse under limited overlap despite high effective sample size (ESS). We introduce Causal Judge Evaluation (CJE), a framework that fixes all three failures. On n=4,961 Chatbot Arena prompts (after filtering from 5k), CJE achieves 99% pairwise ranking accuracy at full sample size (94% averaged across configurations), matching oracle quality, at 14x lower cost (for ranking 5 policies) by calibrating a 16x cheaper judge on just 5% oracle labels (~250 labels). CJE combines three components: (i) AutoCal-R, reward calibration via mean-preserving isotonic regression; (ii) SIMCal-W, weight stabilization via stacking of S-monotone candidates; and (iii) Oracle-Uncertainty Aware (OUA) inference that propagates calibration uncertainty into confidence intervals. We formalize the Coverage-Limited Efficiency (CLE) diagnostic, which explains why IPS-style estimators fail even when ESS exceeds 90%: the logger rarely visits regions where target policies concentrate. Key findings: SNIPS inverts rankings even with reward calibration (38% pairwise, negative Kendall's tau) due to weight instability; calibrated IPS remains near-random (47%) despite weight stabilization, consistent with CLE; OUA improves coverage from near-0% to ~86% (Direct) and ~96% (stacked-DR), where naive intervals severely under-cover.

A well-calibrated model should express confidence that matches its actual accuracy -- when it claims 80\% confidence, it should be correct 80\% of the time. While large language models (LLMs) have achieved remarkable performance across diverse tasks, their epistemic calibration remains poorly understood. We introduce KalshiBench, a benchmark of 300 prediction market questions from Kalshi, a CFTC-regulated exchange, with verifiable real-world outcomes occurring after model training cutoffs. Unlike traditional benchmarks measuring accuracy on static knowledge, KalshiBench evaluates whether models can appropriately quantify uncertainty about genuinely unknown future events. We evaluate five frontier models -- Claude Opus 4.5, GPT-5.2, DeepSeek-V3.2, Qwen3-235B, and Kimi-K2 -- and find systematic overconfidence across all models. Even the best-calibrated model (Claude Opus 4.5, ECE=0.120) shows substantial calibration errors, while reasoning-enhanced models like GPT-5.2-XHigh exhibit worse calibration (ECE=0.395) despite comparable accuracy. Critically, only one model achieves a positive Brier Skill Score, indicating most models perform worse than simply predicting base rates. Our findings suggest that scaling and enhanced reasoning do not automatically confer calibration benefits, highlighting epistemic calibration as a distinct capability requiring targeted development.

Large language models (LLMs) have demonstrated remarkable capabilities out of box for a wide range of applications, yet accuracy still remains a major growth area, especially in mission-critical domains such as biomedicine. An effective method to calibrate the confidence level on LLM responses is essential to automatically detect errors and facilitate human-in-the-loop verification. An important source of calibration signals stems from expert-stipulated programmatic supervision, which is often available at low cost but has its own limitations such as noise and coverage. In this paper, we introduce a Pareto optimal self-supervision framework that can leverage available programmatic supervision to systematically calibrate LLM responses by producing a risk score for every response, without any additional manual efforts. This is accomplished by learning a harmonizer model to align LLM output with other available supervision sources, which would assign higher risk scores to more uncertain LLM responses and facilitate error correction. Experiments on standard relation extraction tasks in biomedical and general domains demonstrate the promise of this approach, with our proposed risk scores highly correlated with the real error rate of LLMs. For the most uncertain test instances, dynamic prompting based on our proposed risk scores results in significant accuracy improvement for off-the-shelf LLMs, boosting GPT-3 results past state-of-the-art (SOTA) weak supervision and GPT-4 results past SOTA supervised results on challenging evaluation datasets.

In this paper, we uncover a systematic bias in the evaluation paradigm of adopting large language models~(LLMs), e.g., GPT-4, as a referee to score and compare the quality of responses generated by candidate models. We find that the quality ranking of candidate responses can be easily hacked by simply altering their order of appearance in the context. This manipulation allows us to skew the evaluation result, making one model appear considerably superior to the other, e.g., Vicuna-13B could beat ChatGPT on 66 over 80 tested queries with ChatGPT as an evaluator. To address this issue, we propose a calibration framework with three simple yet effective strategies: 1) Multiple Evidence Calibration, which requires the evaluator model to generate multiple evaluation evidence before assigning ratings; 2) Balanced Position Calibration, which aggregates results across various orders to determine the final score; 3) Human-in-the-Loop Calibration, which introduces a balanced position diversity entropy to measure the difficulty of each example and seeks human assistance when needed. We also manually annotate the "win/tie/lose" outcomes of responses from ChatGPT and Vicuna-13B in the Vicuna Benchmark's question prompt, and extensive experiments demonstrate that our approach successfully mitigates evaluation bias, resulting in closer alignment with human judgments. We release our code and human annotation at https://github.com/i-Eval/FairEval to facilitate future research.

Whenever we use devices to take measurements, calibration is indispensable. While the purpose of calibration is to reduce bias and uncertainty in the measurements, it can be quite difficult, expensive, and sometimes even impossible to implement. We study a challenging problem called self-calibration, i.e., the task of designing an algorithm for devices so that the algorithm is able to perform calibration automatically. More precisely, we consider the setup y = A(d) x + epsilon where only partial information about the sensing matrix A(d) is known and where A(d) linearly depends on d. The goal is to estimate the calibration parameter d (resolve the uncertainty in the sensing process) and the signal/object of interests x simultaneously. For three different models of practical relevance, we show how such a bilinear inverse problem, including blind deconvolution as an important example, can be solved via a simple linear least squares approach. As a consequence, the proposed algorithms are numerically extremely efficient, thus potentially allowing for real-time deployment. We also present a variation of the least squares approach, which leads to a~spectral method, where the solution to the bilinear inverse problem can be found by computing the singular vector associated with the smallest singular value of a certain matrix derived from the bilinear system. Explicit theoretical guarantees and stability theory are derived for both techniques; and the number of sampling complexity is nearly optimal (up to a poly-log factor). Applications in imaging sciences and signal processing are discussed and numerical simulations are presented to demonstrate the effectiveness and efficiency of our approach.

The fusion of LiDARs and cameras has been increasingly adopted in autonomous driving for perception tasks. The performance of such fusion-based algorithms largely depends on the accuracy of sensor calibration, which is challenging due to the difficulty of identifying common features across different data modalities. Previously, many calibration methods involved specific targets and/or manual intervention, which has proven to be cumbersome and costly. Learning-based online calibration methods have been proposed, but their performance is barely satisfactory in most cases. These methods usually suffer from issues such as sparse feature maps, unreliable cross-modality association, inaccurate calibration parameter regression, etc. In this paper, to address these issues, we propose CalibFormer, an end-to-end network for automatic LiDAR-camera calibration. We aggregate multiple layers of camera and LiDAR image features to achieve high-resolution representations. A multi-head correlation module is utilized to identify correlations between features more accurately. Lastly, we employ transformer architectures to estimate accurate calibration parameters from the correlation information. Our method achieved a mean translation error of 0.8751 cm and a mean rotation error of 0.0562 ^{circ} on the KITTI dataset, surpassing existing state-of-the-art methods and demonstrating strong robustness, accuracy, and generalization capabilities.

Given the importance of getting calibrated predictions and reliable uncertainty estimations, various post-hoc calibration methods have been developed for neural networks on standard multi-class classification tasks. However, these methods are not well suited for calibrating graph neural networks (GNNs), which presents unique challenges such as accounting for the graph structure and the graph-induced correlations between the nodes. In this work, we conduct a systematic study on the calibration qualities of GNN node predictions. In particular, we identify five factors which influence the calibration of GNNs: general under-confident tendency, diversity of nodewise predictive distributions, distance to training nodes, relative confidence level, and neighborhood similarity. Furthermore, based on the insights from this study, we design a novel calibration method named Graph Attention Temperature Scaling (GATS), which is tailored for calibrating graph neural networks. GATS incorporates designs that address all the identified influential factors and produces nodewise temperature scaling using an attention-based architecture. GATS is accuracy-preserving, data-efficient, and expressive at the same time. Our experiments empirically verify the effectiveness of GATS, demonstrating that it can consistently achieve state-of-the-art calibration results on various graph datasets for different GNN backbones.

Extrinsic calibration is essential for multi-sensor fusion, existing methods rely on structured targets or fully-excited data, limiting real-world applicability. Online calibration further suffers from weak excitation, leading to unreliable estimates. To address these limitations, we propose a reinforcement learning (RL)-based extrinsic calibration framework that formulates extrinsic calibration as a decision-making problem, directly optimizes SE(3) extrinsics to enhance odometry accuracy. Our approach leverages a probabilistic Bingham distribution to model 3D rotations, ensuring stable optimization while inherently retaining quaternion symmetry. A trajectory alignment reward mechanism enables robust calibration without structured targets by quantitatively evaluating estimated tightly-coupled trajectory against a reference trajectory. Additionally, an automated data selection module filters uninformative samples, significantly improving efficiency and scalability for large-scale datasets. Extensive experiments on UAVs, UGVs, and handheld platforms demonstrate that our method outperforms traditional optimization-based approaches, achieving high-precision calibration even under weak excitation conditions. Our framework simplifies deployment on diverse robotic platforms by eliminating the need for high-quality initial extrinsics and enabling calibration from routine operating data. The code is available at https://github.com/APRIL-ZJU/learn-to-calibrate.

Deep neural networks are increasingly utilized in various machine learning tasks. However, as these models grow in complexity, they often face calibration issues, despite enhanced prediction accuracy. Many studies have endeavored to improve calibration performance through the use of specific loss functions, data preprocessing and training frameworks. Yet, investigations into calibration properties have been somewhat overlooked. Our study leverages the Neural Architecture Search (NAS) search space, offering an exhaustive model architecture space for thorough calibration properties exploration. We specifically create a model calibration dataset. This dataset evaluates 90 bin-based and 12 additional calibration measurements across 117,702 unique neural networks within the widely employed NATS-Bench search space. Our analysis aims to answer several longstanding questions in the field, using our proposed dataset: (i) Can model calibration be generalized across different datasets? (ii) Can robustness be used as a calibration measurement? (iii) How reliable are calibration metrics? (iv) Does a post-hoc calibration method affect all models uniformly? (v) How does calibration interact with accuracy? (vi) What is the impact of bin size on calibration measurement? (vii) Which architectural designs are beneficial for calibration? Additionally, our study bridges an existing gap by exploring calibration within NAS. By providing this dataset, we enable further research into NAS calibration. As far as we are aware, our research represents the first large-scale investigation into calibration properties and the premier study of calibration issues within NAS. The project page can be found at https://www.taolinwei.com/calibration-study

We propose causal isotonic calibration, a novel nonparametric method for calibrating predictors of heterogeneous treatment effects. Furthermore, we introduce cross-calibration, a data-efficient variant of calibration that eliminates the need for hold-out calibration sets. Cross-calibration leverages cross-fitted predictors and generates a single calibrated predictor using all available data. Under weak conditions that do not assume monotonicity, we establish that both causal isotonic calibration and cross-calibration achieve fast doubly-robust calibration rates, as long as either the propensity score or outcome regression is estimated accurately in a suitable sense. The proposed causal isotonic calibrator can be wrapped around any black-box learning algorithm, providing robust and distribution-free calibration guarantees while preserving predictive performance.

Conformal Prediction (CP) controls the prediction uncertainty of classification systems by producing a small prediction set, ensuring a predetermined probability that the true class lies within this set. This is commonly done by defining a score, based on the model predictions, and setting a threshold on this score using a validation set. In this study, we address the problem of CP calibration when we only have access to a calibration set with noisy labels. We show how we can estimate the noise-free conformal threshold based on the noisy labeled data. We derive a finite sample coverage guarantee for uniform noise that remains effective even in tasks with a large number of classes. We dub our approach Noise-Aware Conformal Prediction (NACP). We illustrate the performance of the proposed results on several standard image classification datasets with a large number of classes.

It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.

When language models (LMs) are trained via reinforcement learning (RL) to generate natural language "reasoning chains", their performance improves on a variety of difficult question answering tasks. Today, almost all successful applications of RL for reasoning use binary reward functions that evaluate the correctness of LM outputs. Because such reward functions do not penalize guessing or low-confidence outputs, they often have the unintended side-effect of degrading calibration and increasing the rate at which LMs generate incorrect responses (or "hallucinate") in other problem domains. This paper describes RLCR (Reinforcement Learning with Calibration Rewards), an approach to training reasoning models that jointly improves accuracy and calibrated confidence estimation. During RLCR, LMs generate both predictions and numerical confidence estimates after reasoning. They are trained to optimize a reward function that augments a binary correctness score with a Brier score -- a scoring rule for confidence estimates that incentivizes calibrated prediction. We first prove that this reward function (or any analogous reward function that uses a bounded, proper scoring rule) yields models whose predictions are both accurate and well-calibrated. We next show that across diverse datasets, RLCR substantially improves calibration with no loss in accuracy, on both in-domain and out-of-domain evaluations -- outperforming both ordinary RL training and classifiers trained to assign post-hoc confidence scores. While ordinary RL hurts calibration, RLCR improves it. Finally, we demonstrate that verbalized confidence can be leveraged at test time to improve accuracy and calibration via confidence-weighted scaling methods. Our results show that explicitly optimizing for calibration can produce more generally reliable reasoning models.

For safety-related applications, it is crucial to produce trustworthy deep neural networks whose prediction is associated with confidence that can represent the likelihood of correctness for subsequent decision-making. Existing dense binary classification models are prone to being over-confident. To improve model calibration, we propose Adaptive Stochastic Label Perturbation (ASLP) which learns a unique label perturbation level for each training image. ASLP employs our proposed Self-Calibrating Binary Cross Entropy (SC-BCE) loss, which unifies label perturbation processes including stochastic approaches (like DisturbLabel), and label smoothing, to correct calibration while maintaining classification rates. ASLP follows Maximum Entropy Inference of classic statistical mechanics to maximise prediction entropy with respect to missing information. It performs this while: (1) preserving classification accuracy on known data as a conservative solution, or (2) specifically improves model calibration degree by minimising the gap between the prediction accuracy and expected confidence of the target training label. Extensive results demonstrate that ASLP can significantly improve calibration degrees of dense binary classification models on both in-distribution and out-of-distribution data. The code is available on https://github.com/Carlisle-Liu/ASLP.

Reliable uncertainty quantification remains a central challenge in predictive modeling. While Bayesian methods are theoretically appealing, their predictive intervals can exhibit poor frequentist calibration, particularly with small sample sizes or model misspecification. We introduce a practical and broadly applicable Bayesian conformal approach based on the influence-function Bayesian bootstrap (BB) with data-driven tuning of the Dirichlet concentration parameter, α. By efficiently approximating the Bayesian bootstrap predictive distribution via influence functions and calibrating α to optimize empirical coverage or average log-probability, our method constructs prediction intervals and distributions that are both well-calibrated and sharp. Across a range of regression models and data settings, this Bayesian conformal framework consistently yields improved empirical coverage and log-score compared to standard Bayesian posteriors. Our procedure is fast, easy to implement, and offers a flexible approach for distributional calibration in predictive modeling.

We present an online post-hoc calibration method, called Online Platt Scaling (OPS), which combines the Platt scaling technique with online logistic regression. We demonstrate that OPS smoothly adapts between i.i.d. and non-i.i.d. settings with distribution drift. Further, in scenarios where the best Platt scaling model is itself miscalibrated, we enhance OPS by incorporating a recently developed technique called calibeating to make it more robust. Theoretically, our resulting OPS+calibeating method is guaranteed to be calibrated for adversarial outcome sequences. Empirically, it is effective on a range of synthetic and real-world datasets, with and without distribution drifts, achieving superior performance without hyperparameter tuning. Finally, we extend all OPS ideas to the beta scaling method.

Contrastively trained encoders have recently been proven to invert the data-generating process: they encode each input, e.g., an image, into the true latent vector that generated the image (Zimmermann et al., 2021). However, real-world observations often have inherent ambiguities. For instance, images may be blurred or only show a 2D view of a 3D object, so multiple latents could have generated them. This makes the true posterior for the latent vector probabilistic with heteroscedastic uncertainty. In this setup, we extend the common InfoNCE objective and encoders to predict latent distributions instead of points. We prove that these distributions recover the correct posteriors of the data-generating process, including its level of aleatoric uncertainty, up to a rotation of the latent space. In addition to providing calibrated uncertainty estimates, these posteriors allow the computation of credible intervals in image retrieval. They comprise images with the same latent as a given query, subject to its uncertainty. Code is available at https://github.com/mkirchhof/Probabilistic_Contrastive_Learning

Machine learning models are widely used, but can also often be wrong. Users would benefit from a reliable indication of whether a given output from a given model should be trusted, so a rational decision can be made whether to use the output or not. For example, outputs can be associated with a confidence measure; if this confidence measure is strongly associated with likelihood of correctness, then the model is said to be well-calibrated. A well-calibrated confidence measure can serve as a basis for rational, graduated decision-making on how much review and care is needed when using generated code. Calibration has so far been studied in mostly non-generative (e.g. classification) settings, especially in software engineering. However, generated code can quite often be wrong: Given generated code, developers must decide whether to use directly, use after varying intensity of careful review, or discard model-generated code. Thus, calibration is vital in generative settings. We make several contributions. We develop a framework for evaluating the calibration of code-generating models. We consider several tasks, correctness criteria, datasets, and approaches, and find that, by and large, generative code models we test are not well-calibrated out of the box. We then show how calibration can be improved using standard methods, such as Platt scaling. Since Platt scaling relies on the prior availability of correctness data, we evaluate the applicability and generalizability of Platt scaling in software engineering, discuss settings where it has good potential for practical use, and settings where it does not. Our contributions will lead to better-calibrated decision-making in the current use of code generated by language models, and offers a framework for future research to further improve calibration methods for generative models in software engineering.

Tool-using agents that act in the world need to be both useful and safe. Well-calibrated model confidences can be used to weigh the risk versus reward of potential actions, but prior work shows that many models are poorly calibrated. Inspired by interpretability literature exploring the internals of models, we propose a novel class of model-internal confidence estimators (MICE) to better assess confidence when calling tools. MICE first decodes from each intermediate layer of the language model using logitLens and then computes similarity scores between each layer's generation and the final output. These features are fed into a learned probabilistic classifier to assess confidence in the decoded output. On the simulated trial and error (STE) tool-calling dataset using Llama3 models, we find that MICE beats or matches the baselines on smoothed expected calibration error. Using MICE confidences to determine whether to call a tool significantly improves over strong baselines on a new metric, expected tool-calling utility. Further experiments show that MICE is sample-efficient, can generalize zero-shot to unseen APIs, and results in higher tool-calling utility in scenarios with varying risk levels. Our code is open source, available at https://github.com/microsoft/mice_for_cats.

High-stakes decision making involves reasoning under uncertainty about the future. In this work, we train language models to make predictions on open-ended forecasting questions. To scale up training data, we synthesize novel forecasting questions from global events reported in daily news, using a fully automated, careful curation recipe. We train the Qwen3 thinking models on our dataset, OpenForesight. To prevent leakage of future information during training and evaluation, we use an offline news corpus, both for data generation and retrieval in our forecasting system. Guided by a small validation set, we show the benefits of retrieval, and an improved reward function for reinforcement learning (RL). Once we obtain our final forecasting system, we perform held-out testing between May to August 2025. Our specialized model, OpenForecaster 8B, matches much larger proprietary models, with our training improving the accuracy, calibration, and consistency of predictions. We find calibration improvements from forecasting training generalize across popular benchmarks. We open-source all our models, code, and data to make research on language model forecasting broadly accessible.

Calibrating deep learning models to yield uncertainty-aware predictions is crucial as deep neural networks get increasingly deployed in safety-critical applications. While existing post-hoc calibration methods achieve impressive results on in-domain test datasets, they are limited by their inability to yield reliable uncertainty estimates in domain-shift and out-of-domain (OOD) scenarios. We aim to bridge this gap by proposing DAC, an accuracy-preserving as well as Density-Aware Calibration method based on k-nearest-neighbors (KNN). In contrast to existing post-hoc methods, we utilize hidden layers of classifiers as a source for uncertainty-related information and study their importance. We show that DAC is a generic method that can readily be combined with state-of-the-art post-hoc methods. DAC boosts the robustness of calibration performance in domain-shift and OOD, while maintaining excellent in-domain predictive uncertainty estimates. We demonstrate that DAC leads to consistently better calibration across a large number of model architectures, datasets, and metrics. Additionally, we show that DAC improves calibration substantially on recent large-scale neural networks pre-trained on vast amounts of data.

Sports betting's recent federal legalisation in the USA coincides with the golden age of machine learning. If bettors can leverage data to reliably predict the probability of an outcome, they can recognise when the bookmaker's odds are in their favour. As sports betting is a multi-billion dollar industry in the USA alone, identifying such opportunities could be extremely lucrative. Many researchers have applied machine learning to the sports outcome prediction problem, generally using accuracy to evaluate the performance of predictive models. We hypothesise that for the sports betting problem, model calibration is more important than accuracy. To test this hypothesis, we train models on NBA data over several seasons and run betting experiments on a single season, using published odds. We show that using calibration, rather than accuracy, as the basis for model selection leads to greater returns, on average (return on investment of +34.69% versus -35.17%) and in the best case (+36.93% versus +5.56%). These findings suggest that for sports betting (or any probabilistic decision-making problem), calibration is a more important metric than accuracy. Sports bettors who wish to increase profits should therefore select their predictive model based on calibration, rather than accuracy.

Popular approaches for quantifying predictive uncertainty in deep neural networks often involve distributions over weights or multiple models, for instance via Markov Chain sampling, ensembling, or Monte Carlo dropout. These techniques usually incur overhead by having to train multiple model instances or do not produce very diverse predictions. This comprehensive and extensive survey aims to familiarize the reader with an alternative class of models based on the concept of Evidential Deep Learning: For unfamiliar data, they aim to admit "what they don't know", and fall back onto a prior belief. Furthermore, they allow uncertainty estimation in a single model and forward pass by parameterizing distributions over distributions. This survey recapitulates existing works, focusing on the implementation in a classification setting, before surveying the application of the same paradigm to regression. We also reflect on the strengths and weaknesses compared to other existing methods and provide the most fundamental derivations using a unified notation to aid future research.

Predictive models often need to work with incomplete information in real-world tasks. Consequently, they must provide reliable probability or confidence estimation, especially in large-scale decision-making and planning tasks. Current large language models (LLMs) are insufficient for accurate estimations, but they can generate relevant factors that may affect the probabilities, produce coarse-grained probabilities when the information is more complete, and help determine which factors are relevant to specific downstream contexts. In this paper, we make use of these capabilities of LLMs to provide a significantly more accurate probabilistic estimation. We propose BIRD, a novel probabilistic inference framework that aligns a Bayesian network with LLM abductions and then estimates more accurate probabilities in a deduction step. We show BIRD provides reliable probability estimations that are 30% better than those provided directly by LLM baselines. These estimates further contribute to better and more trustworthy decision making.

Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty. Predictive uncertainty is commonly measured by the entropy of the Bayesian model average (BMA) predictive distribution. Yet, the properness of this current measure of predictive uncertainty was recently questioned. We provide new insights regarding those limitations. Our analyses show that the current measure erroneously assumes that the BMA predictive distribution is equivalent to the predictive distribution of the true model that generated the dataset. Consequently, we introduce a theoretically grounded measure to overcome these limitations. We experimentally verify the benefits of our introduced measure of predictive uncertainty. We find that our introduced measure behaves more reasonably in controlled synthetic tasks. Moreover, our evaluations on ImageNet demonstrate that our introduced measure is advantageous in real-world applications utilizing predictive uncertainty.

We address the neural network robustness problem by adding Similarity (i.e., correctly predicted depth-matches into training)-awareness and Distance-to-training-distribution-awareness to the existing output Magnitude (i.e., decision-boundary)-awareness of the softmax function. The resulting SDM activation function provides strong signals of the relative epistemic (reducible) predictive uncertainty. We use this novel behavior to further address the complementary HCI problem of mapping the output to human-interpretable summary statistics over relevant partitions of a held-out calibration set. Estimates of prediction-conditional uncertainty are obtained via a parsimonious learned transform over the class-conditional empirical CDFs of the output of a final-layer SDM activation function. For decision-making and as an intrinsic model check, estimates of class-conditional accuracy are obtained by further partitioning the high-probability regions of this calibrated output into class-conditional, region-specific CDFs. The uncertainty estimates from SDM calibration are remarkably robust to test-time distribution shifts and out-of-distribution inputs; incorporate awareness of the effective sample size; provide estimates of uncertainty from the learning and data splitting processes; and are well-suited for selective classification and conditional branching for additional test-time compute based on the predictive uncertainty, as for selective LLM generation, routing, and composition over multiple models and retrieval. Finally, we construct SDM networks, LLMs with uncertainty-aware verification and interpretability-by-exemplar as intrinsic properties. We provide open-source software implementing these results.

Bayesian parameter inference is useful to improve Li-ion battery diagnostics and can help formulate battery aging models. However, it is computationally intensive and cannot be easily repeated for multiple cycles, multiple operating conditions, or multiple replicate cells. To reduce the computational cost of Bayesian calibration, numerical solvers for physics-based models can be replaced with faster surrogates. A physics-informed neural network (PINN) is developed as a surrogate for the pseudo-2D (P2D) battery model calibration. For the P2D surrogate, additional training regularization was needed as compared to the PINN single-particle model (SPM) developed in Part I. Both the PINN SPM and P2D surrogate models are exercised for parameter inference and compared to data obtained from a direct numerical solution of the governing equations. A parameter inference study highlights the ability to use these PINNs to calibrate scaling parameters for the cathode Li diffusion and the anode exchange current density. By realizing computational speed-ups of 2250x for the P2D model, as compared to using standard integrating methods, the PINN surrogates enable rapid state-of-health diagnostics. In the low-data availability scenario, the testing error was estimated to 2mV for the SPM surrogate and 10mV for the P2D surrogate which could be mitigated with additional data.

By design, average precision (AP) for object detection aims to treat all classes independently: AP is computed independently per category and averaged. On one hand, this is desirable as it treats all classes equally. On the other hand, it ignores cross-category confidence calibration, a key property in real-world use cases. Unfortunately, under important conditions (i.e., large vocabulary, high instance counts) the default implementation of AP is neither category independent, nor does it directly reward properly calibrated detectors. In fact, we show that on LVIS the default implementation produces a gameable metric, where a simple, un-intuitive re-ranking policy can improve AP by a large margin. To address these limitations, we introduce two complementary metrics. First, we present a simple fix to the default AP implementation, ensuring that it is independent across categories as originally intended. We benchmark recent LVIS detection advances and find that many reported gains do not translate to improvements under our new evaluation, suggesting recent improvements may arise from difficult to interpret changes to cross-category rankings. Given the importance of reliably benchmarking cross-category rankings, we consider a pooled version of AP (AP-Pool) that rewards properly calibrated detectors by directly comparing cross-category rankings. Finally, we revisit classical approaches for calibration and find that explicitly calibrating detectors improves state-of-the-art on AP-Pool by 1.7 points

We study the connection between multicalibration and boosting for squared error regression. First we prove a useful characterization of multicalibration in terms of a ``swap regret'' like condition on squared error. Using this characterization, we give an exceedingly simple algorithm that can be analyzed both as a boosting algorithm for regression and as a multicalibration algorithm for a class H that makes use only of a standard squared error regression oracle for H. We give a weak learning assumption on H that ensures convergence to Bayes optimality without the need to make any realizability assumptions -- giving us an agnostic boosting algorithm for regression. We then show that our weak learning assumption on H is both necessary and sufficient for multicalibration with respect to H to imply Bayes optimality. We also show that if H satisfies our weak learning condition relative to another class C then multicalibration with respect to H implies multicalibration with respect to C. Finally we investigate the empirical performance of our algorithm experimentally using an open source implementation that we make available. Our code repository can be found at https://github.com/Declancharrison/Level-Set-Boosting.

Computational sensing strategies often suffer from calibration errors in the physical implementation of their ideal sensing models. Such uncertainties are typically addressed by using multiple, accurately chosen training signals to recover the missing information on the sensing model, an approach that can be resource-consuming and cumbersome. Conversely, blind calibration does not employ any training signal, but corresponds to a bilinear inverse problem whose algorithmic solution is an open issue. We here address blind calibration as a non-convex problem for linear random sensing models, in which we aim to recover an unknown signal from its projections on sub-Gaussian random vectors, each subject to an unknown positive multiplicative factor (or gain). To solve this optimisation problem we resort to projected gradient descent starting from a suitable, carefully chosen initialisation point. An analysis of this algorithm allows us to show that it converges to the exact solution provided a sample complexity requirement is met, i.e., relating convergence to the amount of information collected during the sensing process. Interestingly, we show that this requirement grows linearly (up to log factors) in the number of unknowns of the problem. This sample complexity is found both in absence of prior information, as well as when subspace priors are available for both the signal and gains, allowing a further reduction of the number of observations required for our recovery guarantees to hold. Moreover, in the presence of noise we show how our descent algorithm yields a solution whose accuracy degrades gracefully with the amount of noise affecting the measurements. Finally, we present some numerical experiments in an imaging context, where our algorithm allows for a simple solution to blind calibration of the gains in a sensor array.

Numerous crucial tasks in real-world decision-making rely on machine learning algorithms with calibrated uncertainty estimates. However, modern methods often yield overconfident and uncalibrated predictions. Various approaches involve training an ensemble of separate models to quantify the uncertainty related to the model itself, known as epistemic uncertainty. In an explicit implementation, the ensemble approach has high computational cost and high memory requirements. This particular challenge is evident in state-of-the-art neural networks such as transformers, where even a single network is already demanding in terms of compute and memory. Consequently, efforts are made to emulate the ensemble model without actually instantiating separate ensemble members, referred to as implicit ensembling. We introduce LoRA-Ensemble, a parameter-efficient deep ensemble method for self-attention networks, which is based on Low-Rank Adaptation (LoRA). Initially developed for efficient LLM fine-tuning, we extend LoRA to an implicit ensembling approach. By employing a single pre-trained self-attention network with weights shared across all members, we train member-specific low-rank matrices for the attention projections. Our method exhibits superior calibration compared to explicit ensembles and achieves similar or better accuracy across various prediction tasks and datasets.

Despite their widespread applications, Large Language Models (LLMs) often struggle to express uncertainty, posing a challenge for reliable deployment in high stakes and safety critical domains like clinical diagnostics. Existing standard baseline methods such as model logits and elicited probabilities produce overconfident and poorly calibrated estimates. In this work, we propose Approximate Bayesian Computation (ABC), a likelihood-free Bayesian inference, based approach that treats LLMs as a stochastic simulator to infer posterior distributions over predictive probabilities. We evaluate our ABC approach on two clinically relevant benchmarks: a synthetic oral lesion diagnosis dataset and the publicly available GretelAI symptom-to-diagnosis dataset. Compared to standard baselines, our approach improves accuracy by up to 46.9\%, reduces Brier scores by 74.4\%, and enhances calibration as measured by Expected Calibration Error (ECE) and predictive entropy.

This paper presents MathBode, a dynamic diagnostic for mathematical reasoning in large language models (LLMs). Instead of one-shot accuracy, MathBode treats each parametric problem as a system: we drive a single parameter sinusoidally and fit first-harmonic responses of model outputs and exact solutions. This yields interpretable, frequency-resolved metrics -- gain (amplitude tracking) and phase (lag) -- that form Bode-style fingerprints. Across five closed-form families (linear solve, ratio/saturation, compound interest, 2x2 linear systems, similar triangles), the diagnostic surfaces systematic low-pass behavior and growing phase lag that accuracy alone obscures. We compare several models against a symbolic baseline that calibrates the instrument (G approx 1, phi approx 0). Results separate frontier from mid-tier models on dynamics, providing a compact, reproducible protocol that complements standard benchmarks with actionable measurements of reasoning fidelity and consistency. We open-source the dataset and code to enable further research and adoption.

Bayesian inference provides principled uncertainty quantification but is often limited by challenges of prior elicitation, likelihood misspecification, and computational burden. The martingale posterior (MGP, Fong et al., 2023) offers an alternative, replacing prior-likelihood elicitation with a predictive rule - namely, a sequence of one-step-ahead predictive distributions - for forward data generation. The utility of MGPs depends on the choice of predictive rule, yet the literature has offered few compelling examples. Foundation transformers are well-suited here, as their autoregressive generation mirrors this forward simulation and their general-purpose design enables rich predictive modeling. We introduce TabMGP, an MGP built on TabPFN, a transformer foundation model that is currently state-of-the-art for tabular data. TabMGP produces credible sets with near-nominal coverage and often outperforms both existing MGP constructions and standard Bayes.

Accurate uncertainty measurement is a key step to building robust and reliable machine learning systems. Conformal prediction is a distribution-free uncertainty quantification algorithm popular for its ease of implementation, statistical coverage guarantees, and versatility for underlying forecasters. However, existing conformal prediction algorithms for time series are limited to single-step prediction without considering the temporal dependency. In this paper, we propose a Copula Conformal Prediction algorithm for multivariate, multi-step Time Series forecasting, CopulaCPTS. We prove that CopulaCPTS has finite sample validity guarantee. On several synthetic and real-world multivariate time series datasets, we show that CopulaCPTS produces more calibrated and sharp confidence intervals for multi-step prediction tasks than existing techniques.

Retrieval Augmented Generation (RAG) complements the knowledge of Large Language Models (LLMs) by leveraging external information to enhance response accuracy for queries. This approach is widely applied in several fields by taking its advantage of injecting the most up-to-date information, and researchers are focusing on understanding and improving this aspect to unlock the full potential of RAG in such high-stakes applications. However, despite the potential of RAG to address these needs, the mechanisms behind the confidence levels of its outputs remain underexplored, although the confidence of information is very critical in some domains, such as finance, healthcare, and medicine. Our study focuses the impact of RAG on confidence within the medical domain under various configurations and models. We evaluate confidence by treating the model's predicted probability as its output and calculating Expected Calibration Error (ECE) and Adaptive Calibration Error (ACE) scores based on the probabilities and accuracy. In addition, we analyze whether the order of retrieved documents within prompts calibrates the confidence. Our findings reveal large variation in confidence and accuracy depending on the model, settings, and the format of input prompts. These results underscore the necessity of optimizing configurations based on the specific model and conditions.

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

We study whether language models can evaluate the validity of their own claims and predict which questions they will be able to answer correctly. We first show that larger models are well-calibrated on diverse multiple choice and true/false questions when they are provided in the right format. Thus we can approach self-evaluation on open-ended sampling tasks by asking models to first propose answers, and then to evaluate the probability "P(True)" that their answers are correct. We find encouraging performance, calibration, and scaling for P(True) on a diverse array of tasks. Performance at self-evaluation further improves when we allow models to consider many of their own samples before predicting the validity of one specific possibility. Next, we investigate whether models can be trained to predict "P(IK)", the probability that "I know" the answer to a question, without reference to any particular proposed answer. Models perform well at predicting P(IK) and partially generalize across tasks, though they struggle with calibration of P(IK) on new tasks. The predicted P(IK) probabilities also increase appropriately in the presence of relevant source materials in the context, and in the presence of hints towards the solution of mathematical word problems. We hope these observations lay the groundwork for training more honest models, and for investigating how honesty generalizes to cases where models are trained on objectives other than the imitation of human writing.

Recently computer-aided diagnosis has demonstrated promising performance, effectively alleviating the workload of clinicians. However, the inherent sample imbalance among different diseases leads algorithms biased to the majority categories, leading to poor performance for rare categories. Existing works formulated this challenge as a long-tailed problem and attempted to tackle it by decoupling the feature representation and classification. Yet, due to the imbalanced distribution and limited samples from tail classes, these works are prone to biased representation learning and insufficient classifier calibration. To tackle these problems, we propose a new Long-tailed Medical Diagnosis (LMD) framework for balanced medical image classification on long-tailed datasets. In the initial stage, we develop a Relation-aware Representation Learning (RRL) scheme to boost the representation ability by encouraging the encoder to capture intrinsic semantic features through different data augmentations. In the subsequent stage, we propose an Iterative Classifier Calibration (ICC) scheme to calibrate the classifier iteratively. This is achieved by generating a large number of balanced virtual features and fine-tuning the encoder using an Expectation-Maximization manner. The proposed ICC compensates for minority categories to facilitate unbiased classifier optimization while maintaining the diagnostic knowledge in majority classes. Comprehensive experiments on three public long-tailed medical datasets demonstrate that our LMD framework significantly surpasses state-of-the-art approaches. The source code can be accessed at https://github.com/peterlipan/LMD.

As Large Language Models (LLMs) become increasingly computationally complex, developing efficient deployment strategies, such as quantization, becomes crucial. State-of-the-art Post-training Quantization (PTQ) techniques often rely on calibration processes to maintain the accuracy of these models. However, while these calibration techniques can enhance performance in certain domains, they may not be as effective in others. This paper aims to draw attention to robust statistical approaches that can mitigate such issues. We propose a weight-adaptive PTQ method that can be considered a precursor to calibration-based PTQ methods, guiding the quantization process to preserve the distribution of weights by minimizing the Kullback-Leibler divergence between the quantized weights and the originally trained weights. This minimization ensures that the quantized model retains the Shannon information content of the original model to a great extent, guaranteeing robust and efficient deployment across many tasks. As such, our proposed approach can perform on par with most common calibration-based PTQ methods, establishing a new pre-calibration step for further adjusting the quantized weights with calibration. We show that our pre-calibration results achieve the same accuracy as some existing calibration-based PTQ methods on various LLMs.

Quantitative tools are increasingly appealing for decision support in healthcare, driven by the growing capabilities of advanced AI systems. However, understanding the predictive uncertainties surrounding a tool's output is crucial for decision-makers to ensure reliable and transparent decisions. In this paper, we present a case study on pulmonary nodule detection for lung cancer screening, enhancing an advanced detection model with an uncertainty quantification technique called conformal risk control (CRC). We demonstrate that prediction sets with conformal guarantees are attractive measures of predictive uncertainty in the safety-critical healthcare domain, allowing end-users to achieve arbitrary validity by trading off false positives and providing formal statistical guarantees on model performance. Among ground-truth nodules annotated by at least three radiologists, our model achieves a sensitivity that is competitive with that generally achieved by individual radiologists, with a slight increase in false positives. Furthermore, we illustrate the risks of using off-the-shelve prediction models when faced with ontological uncertainty, such as when radiologists disagree on what constitutes the ground truth on pulmonary nodules.

Conformal prediction is a framework for predictive inference with a distribution-free, finite-sample guarantee. However, it tends to provide uninformative prediction sets when calibration data are scarce. This paper introduces Synthetic-powered predictive inference (SPI), a novel framework that incorporates synthetic data -- e.g., from a generative model -- to improve sample efficiency. At the core of our method is a score transporter: an empirical quantile mapping that aligns nonconformity scores from trusted, real data with those from synthetic data. By carefully integrating the score transporter into the calibration process, SPI provably achieves finite-sample coverage guarantees without making any assumptions about the real and synthetic data distributions. When the score distributions are well aligned, SPI yields substantially tighter and more informative prediction sets than standard conformal prediction. Experiments on image classification -- augmenting data with synthetic diffusion-model generated images -- and on tabular regression demonstrate notable improvements in predictive efficiency in data-scarce settings.

Real-world large-scale datasets are both noisily labeled and class-imbalanced. The issues seriously hurt the generalization of trained models. It is hence significant to address the simultaneous incorrect labeling and class-imbalance, i.e., the problem of learning with noisy labels on long-tailed data. Previous works develop several methods for the problem. However, they always rely on strong assumptions that are invalid or hard to be checked in practice. In this paper, to handle the problem and address the limitations of prior works, we propose a representation calibration method RCAL. Specifically, RCAL works with the representations extracted by unsupervised contrastive learning. We assume that without incorrect labeling and class imbalance, the representations of instances in each class conform to a multivariate Gaussian distribution, which is much milder and easier to be checked. Based on the assumption, we recover underlying representation distributions from polluted ones resulting from mislabeled and class-imbalanced data. Additional data points are then sampled from the recovered distributions to help generalization. Moreover, during classifier training, representation learning takes advantage of representation robustness brought by contrastive learning, which further improves the classifier performance. We derive theoretical results to discuss the effectiveness of our representation calibration. Experiments on multiple benchmarks justify our claims and confirm the superiority of the proposed method.

In any ranking system, the retrieval model outputs a single score for a document based on its belief on how relevant it is to a given search query. While retrieval models have continued to improve with the introduction of increasingly complex architectures, few works have investigated a retrieval model's belief in the score beyond the scope of a single value. We argue that capturing the model's uncertainty with respect to its own scoring of a document is a critical aspect of retrieval that allows for greater use of current models across new document distributions, collections, or even improving effectiveness for down-stream tasks. In this paper, we address this problem via an efficient Bayesian framework for retrieval models which captures the model's belief in the relevance score through a stochastic process while adding only negligible computational overhead. We evaluate this belief via a ranking based calibration metric showing that our approximate Bayesian framework significantly improves a retrieval model's ranking effectiveness through a risk aware reranking as well as its confidence calibration. Lastly, we demonstrate that this additional uncertainty information is actionable and reliable on down-stream tasks represented via cutoff prediction.

Post-Training Quantization (PTQ) enhances the efficiency of Large Language Models (LLMs) by enabling faster operation and compatibility with more accessible hardware through reduced memory usage, at the cost of small performance drops. We explore the role of calibration sets in PTQ, specifically their effect on hidden activations in various notable open-source LLMs. Calibration sets are crucial for evaluating activation magnitudes and identifying outliers, which can distort the quantization range and negatively impact performance. Our analysis reveals a marked contrast in quantization effectiveness across models. The older OPT model, upon which much of the quantization literature is based, shows significant performance deterioration and high susceptibility to outliers with varying calibration sets. In contrast, newer models like Llama-2 7B, Llama-3 8B, Command-R 35B, and Mistral 7B demonstrate strong robustness, with Mistral 7B showing near-immunity to outliers and stable activations. These findings suggest a shift in PTQ strategies might be needed. As advancements in pre-training methods reduce the relevance of outliers, there is an emerging need to reassess the fundamentals of current quantization literature. The emphasis should pivot towards optimizing inference speed, rather than primarily focusing on outlier preservation, to align with the evolving characteristics of state-of-the-art LLMs.

The bio-inspired event camera has garnered extensive research attention in recent years, owing to its significant potential derived from its high dynamic range and low latency characteristics. Similar to the standard camera, the event camera requires precise intrinsic calibration to facilitate further high-level visual applications, such as pose estimation and mapping. While several calibration methods for event cameras have been proposed, most of them are either (i) engineering-driven, heavily relying on conventional image-based calibration pipelines, or (ii) inconvenient, requiring complex instrumentation. To this end, we propose an accurate and convenient intrinsic calibration method for event cameras, named eKalibr, which builds upon a carefully designed event-based circle grid pattern recognition algorithm. To extract target patterns from events, we perform event-based normal flow estimation to identify potential events generated by circle edges, and cluster them spatially. Subsequently, event clusters associated with the same grid circles are matched and grouped using normal flows, for subsequent time-varying ellipse estimation. Fitted ellipse centers are time-synchronized, for final grid pattern recognition. We conducted extensive experiments to evaluate the performance of eKalibr in terms of pattern extraction and intrinsic calibration. The implementation of eKalibr is open-sourced at (https://github.com/Unsigned-Long/eKalibr) to benefit the research community.

High-quality calibrated uncertainty estimates are crucial for numerous real-world applications, especially for deep learning-based deployed ML systems. While Bayesian deep learning techniques allow uncertainty estimation, training them with large-scale datasets is an expensive process that does not always yield models competitive with non-Bayesian counterparts. Moreover, many of the high-performing deep learning models that are already trained and deployed are non-Bayesian in nature and do not provide uncertainty estimates. To address these issues, we propose BayesCap that learns a Bayesian identity mapping for the frozen model, allowing uncertainty estimation. BayesCap is a memory-efficient method that can be trained on a small fraction of the original dataset, enhancing pretrained non-Bayesian computer vision models by providing calibrated uncertainty estimates for the predictions without (i) hampering the performance of the model and (ii) the need for expensive retraining the model from scratch. The proposed method is agnostic to various architectures and tasks. We show the efficacy of our method on a wide variety of tasks with a diverse set of architectures, including image super-resolution, deblurring, inpainting, and crucial application such as medical image translation. Moreover, we apply the derived uncertainty estimates to detect out-of-distribution samples in critical scenarios like depth estimation in autonomous driving. Code is available at https://github.com/ExplainableML/BayesCap.

The alignment of large language models (LLMs) with human values increasingly relies on using other LLMs as automated judges, or ``autoraters''. However, their reliability is limited by a foundational issue: they are trained on discrete preference labels, forcing a single ground truth onto tasks that are often subjective, ambiguous, or nuanced. We argue that a reliable autorater must learn to model the full distribution of preferences defined by a target population. In this paper, we propose a general framework for calibrating probabilistic autoraters to any given preference distribution. We formalize the problem and present two learning methods tailored to different data conditions: 1) a direct supervised fine-tuning for dense, probabilistic labels, and 2) a reinforcement learning approach for sparse, binary labels. Our empirical results show that finetuning autoraters with a distribution-matching objective leads to verbalized probability predictions that are better aligned with the target preference distribution, with improved calibration and significantly lower positional bias, all while preserving performance on objective tasks.

Datasets scraped from the internet have been critical to the successes of large-scale machine learning. Yet, this very success puts the utility of future internet-derived datasets at potential risk, as model outputs begin to replace human annotations as a source of supervision. In this work, we first formalize a system where interactions with one model are recorded as history and scraped as training data in the future. We then analyze its stability over time by tracking changes to a test-time bias statistic (e.g. gender bias of model predictions). We find that the degree of bias amplification is closely linked to whether the model's outputs behave like samples from the training distribution, a behavior which we characterize and define as consistent calibration. Experiments in three conditional prediction scenarios - image classification, visual role-labeling, and language generation - demonstrate that models that exhibit a sampling-like behavior are more calibrated and thus more stable. Based on this insight, we propose an intervention to help calibrate and stabilize unstable feedback systems. Code is available at https://github.com/rtaori/data_feedback.

Studying low-likelihood high-impact extreme weather events in a warming world is a significant and challenging task for current ensemble forecasting systems. While these systems presently use up to 100 members, larger ensembles could enrich the sampling of internal variability. They may capture the long tails associated with climate hazards better than traditional ensemble sizes. Due to computational constraints, it is infeasible to generate huge ensembles (comprised of 1,000-10,000 members) with traditional, physics-based numerical models. In this two-part paper, we replace traditional numerical simulations with machine learning (ML) to generate hindcasts of huge ensembles. In Part I, we construct an ensemble weather forecasting system based on Spherical Fourier Neural Operators (SFNO), and we discuss important design decisions for constructing such an ensemble. The ensemble represents model uncertainty through perturbed-parameter techniques, and it represents initial condition uncertainty through bred vectors, which sample the fastest growing modes of the forecast. Using the European Centre for Medium-Range Weather Forecasts Integrated Forecasting System (IFS) as a baseline, we develop an evaluation pipeline composed of mean, spectral, and extreme diagnostics. Using large-scale, distributed SFNOs with 1.1 billion learned parameters, we achieve calibrated probabilistic forecasts. As the trajectories of the individual members diverge, the ML ensemble mean spectra degrade with lead time, consistent with physical expectations. However, the individual ensemble members' spectra stay constant with lead time. Therefore, these members simulate realistic weather states, and the ML ensemble thus passes a crucial spectral test in the literature. The IFS and ML ensembles have similar Extreme Forecast Indices, and we show that the ML extreme weather forecasts are reliable and discriminating.

Quantile regression is a fundamental problem in statistical learning motivated by a need to quantify uncertainty in predictions, or to model a diverse population without being overly reductive. For instance, epidemiological forecasts, cost estimates, and revenue predictions all benefit from being able to quantify the range of possible values accurately. As such, many models have been developed for this problem over many years of research in statistics, machine learning, and related fields. Rather than proposing yet another (new) algorithm for quantile regression we adopt a meta viewpoint: we investigate methods for aggregating any number of conditional quantile models, in order to improve accuracy and robustness. We consider weighted ensembles where weights may vary over not only individual models, but also over quantile levels, and feature values. All of the models we consider in this paper can be fit using modern deep learning toolkits, and hence are widely accessible (from an implementation point of view) and scalable. To improve the accuracy of the predicted quantiles (or equivalently, prediction intervals), we develop tools for ensuring that quantiles remain monotonically ordered, and apply conformal calibration methods. These can be used without any modification of the original library of base models. We also review some basic theory surrounding quantile aggregation and related scoring rules, and contribute a few new results to this literature (for example, the fact that post sorting or post isotonic regression can only improve the weighted interval score). Finally, we provide an extensive suite of empirical comparisons across 34 data sets from two different benchmark repositories.

Some classical uncertainty quantification problems require the estimation of multiple expectations. Estimating all of them accurately is crucial and can have a major impact on the analysis to perform, and standard existing Monte Carlo methods can be costly to do so. We propose here a new procedure based on importance sampling and control variates for estimating more efficiently multiple expectations with the same sample. We first show that there exists a family of optimal estimators combining both importance sampling and control variates, which however cannot be used in practice because they require the knowledge of the values of the expectations to estimate. Motivated by the form of these optimal estimators and some interesting properties, we therefore propose an adaptive algorithm. The general idea is to adaptively update the parameters of the estimators for approaching the optimal ones. We suggest then a quantitative stopping criterion that exploits the trade-off between approaching these optimal parameters and having a sufficient budget left. This left budget is then used to draw a new independent sample from the final sampling distribution, allowing to get unbiased estimators of the expectations. We show how to apply our procedure to sensitivity analysis, by estimating Sobol' indices and quantifying the impact of the input distributions. Finally, realistic test cases show the practical interest of the proposed algorithm, and its significant improvement over estimating the expectations separately.

Large Language Models (LLMs) have demonstrated great capabilities in solving a wide range of tasks in a resource-efficient manner through prompting, which does not require task-specific training, but suffers from performance fluctuation when there are multiple prompt candidates. Previous works have introduced gradient-free probability-based prompt selection methods that aim to choose the optimal prompt among the candidates for a given task but fail to provide a comprehensive and fair comparison between each other. In this paper, we propose a unified framework to interpret and evaluate the existing probability-based prompt selection methods by performing extensive experiments on 13 common NLP tasks. We find that all existing methods can be unified into some variant of the method that maximizes the mutual information between the input and the corresponding model output (denoted as MI). Using the finding, we develop several variants of MI and increases the effectiveness of the best prompt selection method from 87.79% to 94.98%, measured as the ratio of the performance of the selected prompt to that of the optimal oracle prompt. Furthermore, we propose a novel calibration method called Calibration by Marginalization (CBM) that is orthogonal to existing methods and helps increase the prompt selection effectiveness of the best method by 99.44%. The code and datasets used in our work will be released at https://github.com/soheeyang/unified-prompt-selection.

Current question-answering benchmarks predominantly focus on accuracy in realizable prediction tasks. Conditioned on a question and answer-key, does the most likely token match the ground truth? Such benchmarks necessarily fail to evaluate LLMs' ability to quantify ground-truth outcome uncertainty. In this work, we focus on the use of LLMs as risk scores for unrealizable prediction tasks. We introduce folktexts, a software package to systematically generate risk scores using LLMs, and evaluate them against US Census data products. A flexible API enables the use of different prompting schemes, local or web-hosted models, and diverse census columns that can be used to compose custom prediction tasks. We evaluate 17 recent LLMs across five proposed benchmark tasks. We find that zero-shot risk scores produced by multiple-choice question-answering have high predictive signal but are widely miscalibrated. Base models consistently overestimate outcome uncertainty, while instruction-tuned models underestimate uncertainty and produce over-confident risk scores. In fact, instruction-tuning polarizes answer distribution regardless of true underlying data uncertainty. This reveals a general inability of instruction-tuned LLMs to express data uncertainty using multiple-choice answers. A separate experiment using verbalized chat-style risk queries yields substantially improved calibration across instruction-tuned models. These differences in ability to quantify data uncertainty cannot be revealed in realizable settings, and highlight a blind-spot in the current evaluation ecosystem that folktexts covers.

When answering questions, LLMs can convey not only an answer, but a level of confidence about the answer being correct. This includes explicit confidence markers (e.g. giving a numeric score) as well as implicit markers, like an authoritative tone or elaborating with additional knowledge. For LLMs to be trustworthy knowledge sources, the confidence they convey should match their actual expertise; however, most current models tend towards overconfidence. To calibrate both implicit and explicit confidence markers, we introduce a pragmatic, listener-aware finetuning method (LACIE) that models the listener, considering not only whether an answer is right, but whether it will be accepted by a listener. We cast calibration as preference optimization, creating data via a two-agent game, where a speaker model's outputs are judged by a simulated listener. We then finetune three LLMs (Mistral-7B, Llama3-8B, Llama3-70B) with LACIE, and show that the resulting models are better calibrated w.r.t. a simulated listener. Crucially, these trends transfer to human listeners, helping them correctly predict model correctness: we conduct a human evaluation where annotators accept or reject an LLM's answers, finding that training with LACIE results in 47% fewer incorrect answers being accepted while maintaining the same level of acceptance for correct answers. Furthermore, LACIE generalizes to another dataset, resulting in a large increase in truthfulness on TruthfulQA when trained on TriviaQA. Our analysis indicates that LACIE leads to a better confidence separation between correct and incorrect examples. Qualitatively, we find that a LACIE-trained model hedges more and implicitly signals certainty when it is correct by using an authoritative tone or including details. Finally, LACIE finetuning leads to an emergent increase in model abstention (e.g. saying "I don't know") for answers that are likely wrong.

Multimodal large language models (MLLMs) hold considerable promise for applications in healthcare. However, their deployment in safety-critical settings is hindered by two key limitations: (i) sensitivity to prompt design, and (ii) a tendency to generate incorrect responses with high confidence. As clinicians may rely on a model's stated confidence to gauge the reliability of its predictions, it is especially important that when a model expresses high confidence, it is also highly accurate. We introduce Prompt4Trust, the first reinforcement learning (RL) framework for prompt augmentation targeting confidence calibration in MLLMs. A lightweight LLM is trained to produce context-aware auxiliary prompts that guide a downstream task MLLM to generate responses in which the expressed confidence more accurately reflects predictive accuracy. Unlike conventional calibration techniques, Prompt4Trust specifically prioritizes aspects of calibration most critical for safe and trustworthy clinical decision-making. Beyond improvements driven by this clinically motivated calibration objective, our proposed method also improves task accuracy, achieving state-of-the-art medical visual question answering (VQA) performance on the PMC-VQA benchmark, which is composed of multiple-choice questions spanning diverse medical imaging modalities. Moreover, our framework trained with a small downstream task MLLM showed promising zero-shot generalization to larger MLLMs in our experiments, suggesting the potential for scalable calibration without the associated computational costs. This work demonstrates the potential of automated yet human-aligned prompt engineering for improving the the trustworthiness of MLLMs in safety critical settings. Our codebase can be found at https://github.com/xingbpshen/prompt4trust.

The increasing complexity of advanced machine learning models requires innovative approaches to manage computational resources effectively. One such method is the Early Exit strategy, which allows for adaptive computation by providing a mechanism to shorten the processing path for simpler data instances. In this paper, we propose EERO, a new methodology to translate the problem of early exiting to a problem of using multiple classifiers with reject option in order to better select the exiting head for each instance. We calibrate the probabilities of exiting at the different heads using aggregation with exponential weights to guarantee a fixed budget .We consider factors such as Bayesian risk, budget constraints, and head-specific budget consumption. Experimental results, conducted using a ResNet-18 model and a ConvNext architecture on Cifar and ImageNet datasets, demonstrate that our method not only effectively manages budget allocation but also enhances accuracy in overthinking scenarios.

We present a comprehensive evaluation of proprietary and open-weights large language models using the first astronomy-specific benchmarking dataset. This dataset comprises 4,425 multiple-choice questions curated from the Annual Review of Astronomy and Astrophysics, covering a broad range of astrophysical topics. Our analysis examines model performance across various astronomical subfields and assesses response calibration, crucial for potential deployment in research environments. Claude-3.5-Sonnet outperforms competitors by up to 4.6 percentage points, achieving 85.0% accuracy. For proprietary models, we observed a universal reduction in cost every 3-to-12 months to achieve similar score in this particular astronomy benchmark. Open-source models have rapidly improved, with LLaMA-3-70b (80.6%) and Qwen-2-72b (77.7%) now competing with some of the best proprietary models. We identify performance variations across topics, with non-English-focused models generally struggling more in exoplanet-related fields, stellar astrophysics, and instrumentation related questions. These challenges likely stem from less abundant training data, limited historical context, and rapid recent developments in these areas. This pattern is observed across both open-weights and proprietary models, with regional dependencies evident, highlighting the impact of training data diversity on model performance in specialized scientific domains. Top-performing models demonstrate well-calibrated confidence, with correlations above 0.9 between confidence and correctness, though they tend to be slightly underconfident. The development for fast, low-cost inference of open-weights models presents new opportunities for affordable deployment in astronomy. The rapid progress observed suggests that LLM-driven research in astronomy may become feasible in the near future.

Background and Objective: Radiation pneumonitis (RP) is a side effect of thoracic radiation therapy. Recently, Machine learning (ML) models enhanced with radiomic and dosiomic features provide better predictions by incorporating spatial information beyond DVHs. However, to improve the clinical decision process, we propose to use uncertainty quantification (UQ) to improve the confidence in model prediction. This study evaluates the impact of post hoc UQ methods on the discriminative performance and calibration of ML models for RP prediction. Methods: This study evaluated four ML models: logistic regression (LR), support vector machines (SVM), extreme gradient boosting (XGB), and random forest (RF), using radiomic, dosiomic, and dosimetric features to predict RP. We applied UQ methods, including Patt scaling, isotonic regression, Venn-ABERS predictor, and Conformal Prediction, to quantify uncertainty. Model performance was assessed through Area Under the Receiver Operating Characteristic curve (AUROC), Area Under the Precision-Recall Curve (AUPRC), and Adaptive Calibration Error (ACE) using Leave-One-Out Cross-Validation (LOO-CV). Results: UQ methods enhanced predictive performance, particularly for high-certainty predictions, while also improving calibration. Radiomic and dosiomic features increased model accuracy but introduced calibration challenges, especially for non-linear models like XGB and RF. Performance gains from UQ methods were most noticeable at higher certainty thresholds. Conclusion: Integrating UQ into ML models with radiomic and dosiomic features improves both predictive accuracy and calibration, supporting more reliable clinical decision-making. The findings emphasize the value of UQ methods in enhancing applicability of predictive models for RP in healthcare settings.

Accurate uncertainty estimates are essential for deploying deep object detectors in safety-critical systems. The development and evaluation of probabilistic object detectors have been hindered by shortcomings in existing performance measures, which tend to involve arbitrary thresholds or limit the detector's choice of distributions. In this work, we propose to view object detection as a set prediction task where detectors predict the distribution over the set of objects. Using the negative log-likelihood for random finite sets, we present a proper scoring rule for evaluating and training probabilistic object detectors. The proposed method can be applied to existing probabilistic detectors, is free from thresholds, and enables fair comparison between architectures. Three different types of detectors are evaluated on the COCO dataset. Our results indicate that the training of existing detectors is optimized toward non-probabilistic metrics. We hope to encourage the development of new object detectors that can accurately estimate their own uncertainty. Code available at https://github.com/georghess/pmb-nll.

Recently, partial Bayesian neural networks (pBNNs), which only consider a subset of the parameters to be stochastic, were shown to perform competitively with full Bayesian neural networks. However, pBNNs are often multi-modal in the latent-variable space and thus challenging to approximate with parametric models. To address this problem, we propose an efficient sampling-based training strategy, wherein the training of a pBNN is formulated as simulating a Feynman--Kac model. We then describe variations of sequential Monte Carlo samplers that allow us to simultaneously estimate the parameters and the latent posterior distribution of this model at a tractable computational cost. We show on various synthetic and real-world datasets that our proposed training scheme outperforms the state of the art in terms of predictive performance.

Real-world datasets follow an imbalanced distribution, which poses significant challenges in rare-category object detection. Recent studies tackle this problem by developing re-weighting and re-sampling methods, that utilise the class frequencies of the dataset. However, these techniques focus solely on the frequency statistics and ignore the distribution of the classes in image space, missing important information. In contrast to them, we propose FRActal CALibration (FRACAL): a novel post-calibration method for long-tailed object detection. FRACAL devises a logit adjustment method that utilises the fractal dimension to estimate how uniformly classes are distributed in image space. During inference, it uses the fractal dimension to inversely downweight the probabilities of uniformly spaced class predictions achieving balance in two axes: between frequent and rare categories, and between uniformly spaced and sparsely spaced classes. FRACAL is a post-processing method and it does not require any training, also it can be combined with many off-the-shelf models such as one-stage sigmoid detectors and two-stage instance segmentation models. FRACAL boosts the rare class performance by up to 8.6% and surpasses all previous methods on LVIS dataset, while showing good generalisation to other datasets such as COCO, V3Det and OpenImages. We provide the code at https://github.com/kostas1515/FRACAL.

Although appearance-based point-of-gaze (PoG) estimation has improved, the estimators still struggle to generalize across individuals due to personal differences. Therefore, person-specific calibration is required for accurate PoG estimation. However, calibrated PoG estimators are often sensitive to head pose variations. To address this, we investigate the key factors influencing calibrated estimators and explore pose-robust calibration strategies. Specifically, we first construct a benchmark, MobilePoG, which includes facial images from 32 individuals focusing on designated points under either fixed or continuously changing head poses. Using this benchmark, we systematically analyze how the diversity of calibration points and head poses influences estimation accuracy. Our experiments show that introducing a wider range of head poses during calibration improves the estimator's ability to handle pose variation. Building on this insight, we propose a dynamic calibration strategy in which users fixate on calibration points while moving their phones. This strategy naturally introduces head pose variation during a user-friendly and efficient calibration process, ultimately producing a better calibrated PoG estimator that is less sensitive to head pose variations than those using conventional calibration strategies. Codes and datasets are available at our project page.

Anomaly detection methods identify examples that do not follow the expected behaviour, typically in an unsupervised fashion, by assigning real-valued anomaly scores to the examples based on various heuristics. These scores need to be transformed into actual predictions by thresholding, so that the proportion of examples marked as anomalies equals the expected proportion of anomalies, called contamination factor. Unfortunately, there are no good methods for estimating the contamination factor itself. We address this need from a Bayesian perspective, introducing a method for estimating the posterior distribution of the contamination factor of a given unlabeled dataset. We leverage on outputs of several anomaly detectors as a representation that already captures the basic notion of anomalousness and estimate the contamination using a specific mixture formulation. Empirically on 22 datasets, we show that the estimated distribution is well-calibrated and that setting the threshold using the posterior mean improves the anomaly detectors' performance over several alternative methods. All code is publicly available for full reproducibility.

In legal decisions, split votes (SV) occur when judges cannot reach a unanimous decision, posing a difficulty for lawyers who must navigate diverse legal arguments and opinions. In high-stakes domains, understanding the alignment of perceived difficulty between humans and AI systems is crucial to build trust. However, existing NLP calibration methods focus on a classifier's awareness of predictive performance, measured against the human majority class, overlooking inherent human label variation (HLV). This paper explores split votes as naturally observable human disagreement and value pluralism. We collect judges' vote distributions from the European Court of Human Rights (ECHR), and present SV-ECHR, a case outcome classification (COC) dataset with SV information. We build a taxonomy of disagreement with SV-specific subcategories. We further assess the alignment of perceived difficulty between models and humans, as well as confidence- and human-calibration of COC models. We observe limited alignment with the judge vote distribution. To our knowledge, this is the first systematic exploration of calibration to human judgements in legal NLP. Our study underscores the necessity for further research on measuring and enhancing model calibration considering HLV in legal decision tasks.

Class imbalance in binary classification tasks remains a significant challenge in machine learning, often resulting in poor performance on minority classes. This study comprehensively evaluates three widely-used strategies for handling class imbalance: Synthetic Minority Over-sampling Technique (SMOTE), Class Weights tuning, and Decision Threshold Calibration. We compare these methods against a baseline scenario of no-intervention across 15 diverse machine learning models and 30 datasets from various domains, conducting a total of 9,000 experiments. Performance was primarily assessed using the F1-score, although our study also tracked results on additional 9 metrics including F2-score, precision, recall, Brier-score, PR-AUC, and AUC. Our results indicate that all three strategies generally outperform the baseline, with Decision Threshold Calibration emerging as the most consistently effective technique. However, we observed substantial variability in the best-performing method across datasets, highlighting the importance of testing multiple approaches for specific problems. This study provides valuable insights for practitioners dealing with imbalanced datasets and emphasizes the need for dataset-specific analysis in evaluating class imbalance handling techniques.

Deep clustering has exhibited remarkable performance; however, the overconfidence problem, i.e., the estimated confidence for a sample belonging to a particular cluster greatly exceeds its actual prediction accuracy, has been overlooked in prior research. To tackle this critical issue, we pioneer the development of a calibrated deep clustering framework. Specifically, we propose a novel dual-head deep clustering pipeline that can effectively calibrate the estimated confidence and the actual accuracy. The calibration head adjusts the overconfident predictions of the clustering head using regularization methods, generating prediction confidence and pseudo-labels that match the model learning status. This calibration process also guides the clustering head in dynamically selecting reliable high-confidence samples for training. Additionally, we introduce an effective network initialization strategy that enhances both training speed and network robustness. Extensive experiments demonstrate the proposed calibrated deep clustering framework not only surpasses state-of-the-art deep clustering methods by approximately 10 times in terms of expected calibration error but also significantly outperforms them in terms of clustering accuracy.

Does prompting a large language model (LLM) like GPT-3 with explanations improve in-context learning? We study this question on two NLP tasks that involve reasoning over text, namely question answering and natural language inference. We test the performance of four LLMs on three textual reasoning datasets using prompts that include explanations in multiple different styles. For these tasks, we find that including explanations in the prompts for OPT, GPT-3 (davinci), and InstructGPT (text-davinci-001) only yields small to moderate accuracy improvements over standard few-show learning. However, text-davinci-002 is able to benefit more substantially. We further show that explanations generated by the LLMs may not entail the models' predictions nor be factually grounded in the input, even on simple tasks with extractive explanations. However, these flawed explanations can still be useful as a way to verify LLMs' predictions post-hoc. Through analysis in our three settings, we show that explanations judged by humans to be good--logically consistent with the input and the prediction--more likely cooccur with accurate predictions. Following these observations, we train calibrators using automatically extracted scores that assess the reliability of explanations, allowing us to improve performance post-hoc across all of our datasets.

Over the last three decades, ensemble forecasts have become an integral part of forecasting the weather. They provide users with more complete information than single forecasts as they permit to estimate the probability of weather events by representing the sources of uncertainties and accounting for the day-to-day variability of error growth in the atmosphere. This paper presents a novel approach to obtain a weather forecast model for ensemble forecasting with machine-learning. AIFS-CRPS is a variant of the Artificial Intelligence Forecasting System (AIFS) developed at ECMWF. Its loss function is based on a proper score, the Continuous Ranked Probability Score (CRPS). For the loss, the almost fair CRPS is introduced because it approximately removes the bias in the score due to finite ensemble size yet avoids a degeneracy of the fair CRPS. The trained model is stochastic and can generate as many exchangeable members as desired and computationally feasible in inference. For medium-range forecasts AIFS-CRPS outperforms the physics-based Integrated Forecasting System (IFS) ensemble for the majority of variables and lead times. For subseasonal forecasts, AIFS-CRPS outperforms the IFS ensemble before calibration and is competitive with the IFS ensemble when forecasts are evaluated as anomalies to remove the influence of model biases.

A critical component in the trustworthiness of LLMs is reliable uncertainty communication, yet LLMs often use assertive language when conveying false claims, leading to over-reliance and eroded trust. We present the first systematic study of faithful confidence calibration of LLMs, benchmarking models' ability to use linguistic expressions of uncertainty that faithfully reflect their intrinsic uncertainty, across a comprehensive array of models, datasets, and prompting strategies. Our results demonstrate that LLMs largely fail at this task, and that existing interventions are insufficient: standard prompt approaches provide only marginal gains, and existing, factuality-based calibration techniques can even harm faithful calibration. To address this critical gap, we introduce MetaFaith, a novel prompt-based calibration approach inspired by human metacognition. We show that MetaFaith robustly improves faithful calibration across diverse models and task domains, enabling up to 61% improvement in faithfulness and achieving an 83% win rate over original generations as judged by humans.