Probability Calibration

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Overview

Probability calibration studies whether predictions made with a given confidence are correct at the corresponding rate, a property that matters when decisions depend on estimated risk rather than just the most likely class. Common approaches include post-hoc methods such as temperature scaling and isotonic regression, recalibration of full predictive distributions, and online or group-conditional methods that seek reliability across changing sequences or subpopulations. Researchers assess calibration with reliability diagnostics and proper scoring rules, and apply these ideas to classifiers, forecasts, medical systems, and language-model agents. A central challenge is that good average calibration may hide subgroup or distribution-shift failures, while improving calibration can interact with accuracy, ranking, and the costs of downstream decisions.

Latest papers

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136 papers32 papers in the last four weeks, up 60% on the four weeks before. 0.2% of all new papers.

Mar 9, 2026

Towards Reliable Simulation-based Inference

Scientific knowledge expands by observing the world, hypothesizing some theories about it, and testing them against collected data. When those theories take the form of statistical models, statistical analyses are involved in the process of testing and refining scientific hypotheses. In this thesis, we focus on statistical models that take the form of scientific simulators and provide background about how machine learning can be used for statistical analyses in this context. The first part of this thesis is about showing empirically that performing statistical analyses with machine learning involves a degree of approximation. Specifically, all statistical analyses involve a level of uncertainty in the conclusions drawn, and we show that approximations can lead to overconfident conclusions. We draw caution regarding such overconfident conclusions and introduce a criterion to diagnose overconfident approximations. In the second part, we introduce balancing, a way to regularize machine learning models to reduce overconfidence and favor calibrated or underconfident approximations. Balancing is first introduced for neural ratio estimation algorithms and then extended to other algorithms. Intuition about why balancing leads to less overconfident solutions is provided, and it is shown empirically that balanced algorithms are often either close to calibrated or underconfident. The third part shows that Bayesian neural networks can also be used to mitigate the overconfidence of approximations. Unlike balancing, no regularization is required, and this solution can then work with few training samples and, hence, computationally expensive simulators. To that end, a new Bayesian neural network prior tailored for simulation-based inference is developed, and empirical results show a reduction in overconfidence compared to similar solutions without Bayesian neural networks.

Arnaud Delaunoy

Department of Electrical Engineering and Computer Science, Faculty of Applied Sciences

Simulation-Based InferenceProbability CalibrationBayesian Neural NetworksDensity Ratio Estimation
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Mar 9, 2026

Micro-Diffusion Compression - Binary Tree Tweedie Denoising for Online Probability Estimation

We present Midicoth, a lossless compression system that introduces a micro-diffusion denoising layer for improving probability estimates produced by adaptive statistical models. In compressors such as Prediction by Partial Matching (PPM), probability estimates are smoothed by a prior to handle sparse observations. When contexts have been seen only a few times, this prior dominates the prediction and produces distributions that are significantly flatter than the true source distribution, leading to compression inefficiency. Midicoth addresses this limitation by treating prior smoothing as a shrinkage process and applying a reverse denoising step that corrects predicted probabilities using empirical calibration statistics. To make this correction data-efficient, the method decomposes each byte prediction into a hierarchy of binary decisions along a bitwise tree. This converts a single 256-way calibration problem into a sequence of binary calibration tasks, enabling reliable estimation of correction terms from relatively small numbers of observations. The denoising process is applied in multiple successive steps, allowing each stage to refine residual prediction errors left by the previous one. The micro-diffusion layer operates as a lightweight post-blend calibration stage applied after all model predictions have been combined, allowing it to correct systematic biases in the final probability distribution. Midicoth combines five fully online components: an adaptive PPM model, a long-range match model, a trie-based word model, a high-order context model, and the micro-diffusion denoiser applied as the final stage.

Roberto Tacconelli

Independent Researcher

Lossless Data CompressionProbability CalibrationDensity Estimation
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Mar 8, 2026

MedQ-Deg: A Multidimensional Benchmark for Evaluating MLLMs Across Medical Image Quality Degradations

Despite impressive performance on standard benchmarks, multimodal large language models (MLLMs) face critical challenges in real-world clinical environments where medical images inevitably suffer various quality degradations. Existing benchmarks exhibit two key limitations: (1) absence of large-scale, multidimensional assessment across medical image quality gradients and (2) no systematic confidence calibration analysis. To address these gaps, we present MedQ-Deg, a comprehensive benchmark for evaluating medical MLLMs under image quality degradations. MedQ-Deg provides multi-dimensional evaluation spanning 18 distinct degradation types, 30 fine-grained capability dimensions, and 7 imaging modalities, with 24,894 question-answer pairs. Each degradation is implemented at 3 severity degrees, calibrated by expert radiologists. We further introduce Calibration Shift metric, which quantifies the gap between a model's perceived confidence and actual performance to assess metacognitive reliability under degradation. Our comprehensive evaluation of 40 mainstream MLLMs reveals several critical findings: (1) overall model performance degrades systematically as degradation severity increases, (2) models universally exhibit the AI Dunning-Kruger Effect, maintaining inappropriately high confidence despite severe accuracy collapse, and (3) models display markedly differentiated behavioral patterns across capability dimensions, imaging modalities, and degradation types. We hope MedQ-Deg drives progress toward medical MLLMs that are robust and trustworthy in real clinical practice.

Jiyao Liu, Junzhi Ning, Chenglong Ma +14

Fudan University, Shanghai, China · Shanghai AI Lab, Shanghai, China · Shanghai Jiaotong University, Shanghai, China +1

MLLM EvaluationMedical VLMsMedical Image BenchmarksMultimodal Model EvaluationImage Corruption RobustnessProbability Calibration
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Mar 8, 2026

Discrete Tokenization Unlocks Transformers for Calibrated Tabular Forecasting

Gradient boosting still dominates Transformers on tabular benchmarks. Our tokenizer uses a deliberately simplistic discretized vocabulary so we can highlight how even basic tokenization unlocks the power of attention on tabular features, yet it already outperforms tuned gradient boosting when combined with Gaussian smoothing. Our solution discretizes environmental context while smoothing labels with adaptive Gaussians, yielding calibrated PDFs. On 600K entities (5M training examples) we outperform tuned XGBoost by 10.8% (35.94s vs 40.31s median MAE) and achieve KS=0.0045 with the adaptive-sigma checkpoint selected to minimize KS rather than median MAE. Ablations confirm architecture matters: losing sequential ordering costs about 2.0%, dropping the time-delta tokens costs about 1.8%, and a stratified calibration analysis reveals where miscalibration persists.

Yael S. Elmatad

TransformerProbability Calibration
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Mar 4, 2026

An intuitive rearranging of the Yates covariance decomposition for probabilistic verification of forecasts with the Brier score

Proper scoring rules are essential for evaluating probabilistic forecasts. We propose a simple algebraic rearrangement of the Yates covariance decomposition of the Brier score into three independently non-negative terms: a variance mismatch term, a correlation deficit term, and a calibration-in-the-large term. This rearrangement makes the optimality conditions for perfect forecasting transparent: the optimal forecast must simultaneously match the variance of outcomes, achieve perfect positive correlation with outcomes, and match the mean of outcomes. Any deviation from these conditions results in a positive contribution to the Brier score.

Bruno Hebling Vieira

Methods of Plasticity Research, Department of Psychology, University of Zurich, Zurich, Switzerland

Probability CalibrationProbabilistic ForecastingProper Scoring Rules
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Feb 27, 2026

V-ECE: Estimating General Expected Calibration Errors

In probabilistic classification, calibration error (CE) measures the average divergence of predicted probabilities f(X)f(X) from P(Y∣f(X))\mathbb{P}(Y|f(X)), the true class distribution for that predicted probability. While being a useful diagnostic tool, it is hard to estimate: popular binning-based estimators are often inconsistent and scale poorly beyond two classes. Recent work rewrites the CE as the excess risk of a model compared to the best recalibration of its own predictions, measured with a proper loss. However, this only works for Bregman-divergence-based calibration errors like the squared error, excluding the more popular L1L_1-distance-based CE. We show that using prediction-dependent proper scores can alleviate this restriction, allowing us to estimate CEs with general convex divergences, including LpL_p distances with closed-form losses in the binary and multiclass settings. To estimate the excess risk, we introduce a more accurate recalibrator that fits a residual to temperature scaling with gradient boosting. The resulting variational estimator, V-ECE, needs no bins or clusters and lower-bounds the true calibration error in expectation. On a benchmark of semi-synthetic tasks built from real classifiers, with known true CE, V-ECE is among the most accurate binary estimators for every calibration error and significantly outperforms all multiclass estimators. Our results are accompanied by additional theory on LpL_p CE, estimator bias, and over- or under-confidence estimation.

Eugène Berta, Sacha Braun, Francis Bach +2

INRIA · Ecole Normale Sup´erieure, PSL Research University · UC Berkeley

Probability CalibrationUncertainty CalibrationPost-Hoc Calibration
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Feb 27, 2026

Tipping the Balance: Impact of Class Imbalance Correction on the Performance of Clinical Risk Prediction Models

Objective: ML-based clinical risk prediction models are increasingly used to support decision-making in healthcare. While class-imbalance correction techniques are commonly applied to improve model performance in settings with rare outcomes, their impact on probabilistic calibration remains insufficiently understood. This study evaluated the effect of widely used resampling strategies on both discrimination and calibration across real-world clinical prediction tasks. Methods: Ten clinical datasets spanning diverse medical domains and including 605,842 patients were analyzed. Multiple machine-learning model families, including linear models and several non-linear approaches, were evaluated. Models were trained on the original data and under three commonly used 1:1 class-imbalance correction strategies (SMOTE, RUS, ROS). Performance was assessed on held-out data using discrimination and calibration metrics. Results: Across all datasets and model families, resampling had no positive impact on predictive performance. Changes in the Receiver Operating Characteristic Area Under Curve (ROC-AUC) relative to models trained on the original data were small and inconsistent (ROS: -0.002, p<0.05; RUS: -0.004, p>0.05; SMOTE: -0.01, p<0.05), with no resampling strategy demonstrating a systematic improvement. In contrast, resampling in general degraded the calibration performance. Models trained using imbalance correction exhibited higher Brier scores (0.029 to 0.080, p<0.05), reflecting poorer probabilistic accuracy, and marked deviations in calibration intercept and slope, indicating systematic distortions of predicted risk despite preserved rank-based performance. Conclusion: In a diverse set of real-world clinical prediction tasks, commonly used class-imbalance correction techniques did not provide generalizable improvements in discrimination and were associated with degraded calibration.

Amalie Koch Andersen, Hadi Mehdizavareh, Arijit Khan +8

Medical Informatics and Image Analysis, Department of Health Science and Technology, Aalborg University, Denmark. · Medical & Translational Science, Novo Nordisk A/S, Søborg, Denmark · Department of Computer Science, Aalborg University, Aalborg, Denmark +10

Class-Imbalanced LearningProbability CalibrationClinical Risk Prediction
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Feb 25, 2026

Epistemic Filtering and Collective Hallucination: A Jury Theorem for Confidence-Calibrated Agents

We investigate the collective accuracy of heterogeneous agents who learn to estimate their own reliability over time and selectively abstain from voting. While classical epistemic voting results, such as the Condorcet Jury Theorem (CJT), assume fixed participation, real-world aggregation often benefits from allowing agents to say ``I don't know.'' We propose a probabilistic framework where agents engage in a calibration phase, updating beliefs about their own fixed competence, before facing a final confidence gate that determines whether to vote or abstain. We derive a non-asymptotic lower bound on the group's success probability and prove that this selective participation generalizes the asymptotic guarantees of the CJT to a sequential, confidence-gated setting. Empirically, we validate these bounds via Monte Carlo simulations. While our results are general, we discuss their potential application to AI safety, outlining how this framework can mitigate hallucinations in collective LLM decision-making.

Jonas Karge

Technische Universität Dresden

Social Choice TheoryAI Agent ReliabilityMulti-Agent ConsensusProbability Calibration
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Feb 20, 2026

Multiclass Calibration Assessment and Recalibration of Probability Predictions via the Linear Log Odds Calibration Function

Machine-generated probability predictions are essential in modern classification tasks such as image classification. A model is well calibrated when its predicted probabilities correspond to observed event frequencies. Despite the need for multicategory recalibration methods, existing methods are limited to (i) comparing calibration between two or more models rather than directly assessing the calibration of a single model, (ii) requiring under-the-hood model access, e.g., accessing logit-scale predictions within the layers of a neural network, and (iii) providing output which is difficult for human analysts to understand. To overcome (i)-(iii), we propose Multicategory Linear Log Odds (MCLLO) recalibration, which (i) includes a likelihood ratio hypothesis test to assess calibration, (ii) does not require under-the-hood access to models and is thus applicable on a wide range of classification problems, and (iii) can be easily interpreted. We demonstrate the effectiveness of the MCLLO method through simulations and three real-world case studies involving image classification via convolutional neural network, obesity analysis via random forest, and ecology via regression modeling. We compare MCLLO to four comparator recalibration techniques utilizing both our hypothesis test and the existing calibration metric Expected Calibration Error to show that our method works well alone and in concert with other methods.

Amy Vennos, Xin Xing, Christopher T. Franck

Department of Statistics, Virginia Polytechnic Institute and State University

Probability CalibrationModel CalibrationPost-Hoc Calibration
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Nov 24, 2025

Diagnosing Corruption-Induced Reliability Failures in Vision-Language Models

Visual corruptions can change vision--language model (VLM) behavior in ways that top-1 accuracy does not capture. A model may keep the same answer while losing distributional support, or improve accuracy through unstable wrong-to-correct changes. We introduce Bench-C, a controlled multiple-choice testbed for studying these effects. It selects semantically diverse samples whose predictions respond to corruption, and evaluates them under 19 corruption types and five severity levels. To measure how corruption changes the option distribution, we introduce the Robustness Alignment Score (RAS), which combines confidence-correctness alignment with uncertainty direction. We further separate originally correct samples from originally wrong samples, and track whether changes are temporary or persistent across severity. Experiments across 13 VLMs reveal a counterintuitive pattern: mild corruptions can improve top-1 accuracy while degrading prediction structure. These failures include silent degradation, erroneous overconfidence, and severity-dependent persistence. Bench-C therefore supports robustness evaluation that goes beyond final answers and attributes where reliability changes occur. Code and data are available at https://github.com/xiangjieSui/Bench-C.

Xiangjie Sui, Songyang Li, Hanwei Zhu +3

City University of Macau, Macau SAR, China · Jiangxi University of Finance and Economics, Nanchang, China · Nanyang Technological University, Singapore

VLM EvaluationVLM RobustnessProbability CalibrationImage Corruption RobustnessNeural Network RobustnessCorruption Robustness
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Sep 17, 2025

Risk-Calibrated Bayesian Streaming Intrusion Detection with SRE-Aligned Decisions

[Corrected v2: an audit found that the score, threshold, and latency descriptions below are not what the shared codebase implements, and that the evaluation streams are assembled constructions. See the correction note on the title page and the corrected companion work, arXiv:2605.24696 (corrected v3), artifact doi:10.5281/zenodo.22673735.] We present a risk-calibrated approach to streaming intrusion detection that couples Bayesian Online Changepoint Detection (BOCPD) with decision thresholds aligned to Site Reliability Engineering (SRE) error budgets. BOCPD provides run-length posteriors that adapt to distribution shift and concept drift; we map these posteriors to alert decisions by optimizing expected operational cost under false-positive and false-negative budgets. We detail the hazard model, conjugate updates, and an O(1)-per-event implementation. A concrete SRE example shows how a 99.9% availability SLO (43.2 minutes per month error budget) yields a probability threshold near 0.91 when missed incidents are 10x more costly than false alarms. We evaluate on the full UNSW-NB15 and CIC-IDS2017 benchmarks with chronological splits, comparing against strong unsupervised baselines (ECOD, COPOD, and LOF). Metrics include PR-AUC, ROC-AUC, Brier score, calibration reliability diagrams, and detection latency measured in events. Results indicate improved precision-recall at mid to high recall and better probability calibration relative to baselines. We release implementation details, hyperparameters, and ablations for hazard sensitivity and computational footprint. Code and reproducibility materials will be made available upon publication; datasets and implementation are available from the corresponding author upon reasonable request.

Michel A. Youssef

Independent Researcher · Independent Researcher, Lebanon

Cost-Sensitive LearningProbability CalibrationOnline Anomaly DetectionNetwork Intrusion Detection
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May 28, 2025

On the Reliability and Stability of Selective Methods in Malware Classification Tasks

The performance figures of modern drift-adaptive malware classifiers appear promising, but does this translate to genuine operational reliability? The standard evaluation paradigm primarily focuses on baseline performance metrics, neglecting confidence-error alignment and operational stability. While prior works established the importance of temporal evaluation and introduced selective classification in malware classification tasks, we take a complementary direction by investigating whether malware classifiers maintain reliable and stable confidence estimates under distribution shifts and exploring the tensions between scientific advancement and practical impacts when they do not. We propose Aurora, a framework to evaluate malware classifiers based on their confidence quality and operational resilience. Aurora subjects the confidence profile of a given model to verification to assess the reliability of its estimates. Unreliable confidence estimates erode operational trust, waste valuable annotation budgets on non-informative samples for active learning, and leave error-prone instances undetected in selective classification. Aurora is further complemented by a set of metrics designed to go beyond point-in-time performance, striving towards a more holistic assessment of operational stability throughout temporal evaluation periods. The fragility we observe in SOTA frameworks across datasets of varying drift severity suggests it may be time to revisit the underlying assumptions.

Alexander Herzog, Aliai Eusebi, Lorenzo Cavallaro

University College London and core64 London, United Kingdom · University College London London, United Kingdom

Malware ClassificationSelective PredictionProbability CalibrationConcept Drift
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Feb 21, 2025

Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling

We study the fundamental problem of calibrating a linear binary classifier of the form σ(w^⊤x)σ(\hat{w}^\top x), where the feature vector xx is Gaussian, σσ is a link function, and w^\hat{w} is an estimator of the true linear weight w⋆w^\star. By interpolating with a noninformative chance classifier\textit{chance classifier}, we construct a well-calibrated predictor whose interpolation weight depends on the angle ∠(w^,w⋆)\angle(\hat{w}, w_\star) between the estimator w^\hat{w} and the true linear weight w⋆w_\star. We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle ∠(w^,w⋆)\angle(\hat{w}, w_\star) can be consistently estimated. Furthermore, the resulting predictor is uniquely Bregman-optimal\textit{Bregman-optimal}, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.

Yufan Li, Pragya Sur

Department of Statistics Harvard University Cambrdige, MA 02138

Probability CalibrationBinary ClassificationStatistical Learning Theory
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Jun 19, 2024

Breaking the T2/3T^{2/3} Barrier for Sequential Calibration

A set of probabilistic forecasts is calibrated if each prediction of the forecaster closely approximates the empirical distribution of outcomes on the subset of timesteps where that prediction was made. We study the fundamental problem of online calibrated forecasting of binary sequences under the standard ℓ1\ell_1 calibration error metric, which was initially studied by Foster & Vohra (1998). They derived an algorithm with O(T2/3)O(T^{2/3}) calibration error after TT time steps, and showed a lower bound of Ω(T1/2)Ω(T^{1/2}). These bounds remained stagnant for two decades, until Qiao & Valiant (2021) improved the lower bound to Ω(T0.528)Ω(T^{0.528}) by introducing a combinatorial game called sign preservation and showing that lower bounds for this game imply lower bounds for calibration. In this paper, we give the first improvement to the O(T2/3)O(T^{2/3}) upper bound on calibration error of Foster & Vohra. We do this by introducing a variant of Qiao & Valiant's game that we call sign preservation with reuse (SPR). We prove that the relationship between SPR and calibrated forecasting is bidirectional: not only do lower bounds for SPR translate into lower bounds for calibration, but algorithms for SPR also translate into new algorithms for calibrated forecasting. We then give an improved upper bound for the SPR game, which implies, via our equivalence, a forecasting algorithm with calibration error O(T2/3−ε)O(T^{2/3 - \varepsilon}) for some ε>0\varepsilon > 0, improving Foster & Vohra's upper bound for the first time. Using similar ideas, we then prove a slightly stronger lower bound than that of Qiao & Valiant, namely Ω(T0.54389)Ω(T^{0.54389}). Our lower bound is obtained by an oblivious adversary, marking the first ω(T1/2)ω(T^{1/2}) calibration lower bound for oblivious adversaries.

Yuval Dagan, Constantinos Daskalakis, Maxwell Fishelson +3

Tel Aviv University · MIT & Archimedes AI · MIT +1

Probability Calibration
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Feb 21, 2023

A probability calibrator for machine-learning classifiers

In fields such as medicine and drug discovery, the ultimate goal of a classification is not to guess a class, but to choose the optimal course of action among a set of possible ones, usually not in one-one correspondence with the set of classes. This decision-theoretic problem requires proper probabilities for the classes. Proper probabilities conditional on the features are computationally almost impossible to find in many important cases. The main idea of the present work is to calculate probabilities conditional not on the features, but on the trained classifier's output. This calculation is cheap, needs to be made only once, and provides an output-to-probability "calibrator" or "transducer" that can be applied to all future outputs of the classifier. In conjunction with problem-dependent utilities, the probabilities of the transducer allow us to find the optimal choice among the classes or among a set of more general decisions, by means of expected-utility maximization. This idea is demonstrated in a simplified drug-discovery problem with a highly imbalanced dataset. The transducer and utility maximization together always lead to improved results, sometimes close to theoretical maximum, for all sets of problem-dependent utilities. The one-time-only calculation of the transducer also provides, automatically: (i) a quantification of the uncertainty about the transducer itself; (ii) the expected utility of the augmented algorithm (including its uncertainty), which can be used for algorithm selection; (iii) the possibility of using the algorithm in a "generative mode", useful if the training dataset is biased.

K. Dyrland, A. S. Lundervold, P. G. L. Porta Mana

Dept of Computer science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, Bergen, Norway · †& Mohn Medical Imaging and Visualization Centre, Dept of Radiology, Haukeland University Hospital, Bergen, Norway

Probability Calibration
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Nov 20, 2022

MetaMax: Improved Open-Set Deep Neural Networks via Weibull Calibration

Open-set recognition refers to the problem in which classes that were not seen during training appear at inference time. This requires the ability to identify instances of novel classes while maintaining discriminative capability for closed-set classification. OpenMax was the first deep neural network-based approach to address open-set recognition by calibrating the predictive scores of a standard closed-set classification network. In this paper we present MetaMax, a more effective post-processing technique that improves upon contemporary methods by directly modeling class activation vectors. MetaMax removes the need for computing class mean activation vectors (MAVs) and distances between a query image and a class MAV as required in OpenMax. Experimental results show that MetaMax outperforms OpenMax and is comparable in performance to other state-of-the-art approaches.

Zongyao Lyu, Nolan B. Gutierrez, William J. Beksi

The University of Texas at Arlington Arlington, TX, USA

Probability CalibrationOpen-Set RecognitionModel CalibrationClassificationImage Classification
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