Conformal Prediction

Conformal prediction is a distribution-free, model-agnostic framework for producing statistically valid prediction sets or intervals that are guaranteed to contain the true outcome with a user-specified probability, using only the mild assumption of exchangeability over the data. A held-out calibration set and a nonconformity score function are sufficient to construct finite-sample marginal coverage guarantees for any base predictor — ranging from linear models and neural networks to Gaussian processes and large language models. The framework unifies classification (prediction sets), regression (prediction intervals), and sequence generation (conformal risk control), and has become a cornerstone method for trustworthy AI deployment in safety-critical and regulated domains.

Semantic Classification

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About

Conformal prediction was introduced by Vladimir Vovk, Alexander Gammerman, and Glenn Shafer in the monograph Algorithmic Learning in a Random World (2005), formalising earlier transductive confidence machine work from the late 1990s. The framework operates under the Exchangeability assumption — a weaker condition than i.i.d., satisfied whenever there is no ordering structure that distinguishes training from test data — and delivers marginal coverage: the probability that the true label or value lies within the prediction set is at least 1 − α for a user-specified error rate α. This guarantee holds in finite samples, for any base model, without any distributional assumptions on the data-generating process. The fundamental result — a consequence of the empirical CDF of exchangeable random variables — is that if one appends any test point y* to an exchangeable calibration set and ranks its nonconformity score among the augmented set, the rank is uniformly distributed, giving the exact coverage property. This elegant probabilistic argument requires no assumptions beyond exchangeability and holds for every base predictor from logistic regression to large Neural Network architectures and foundation models.

The core algorithmic protocol (full conformal) fits the model on all n training points, computes a Nonconformity Score for each (for regression, a common choice is |y − ŷ|; for classification, 1 minus the softmax probability of the true class), and selects the ⌈(1 − α)(n + 1)⌉-th order statistic of calibration nonconformity scores as the threshold τ. At test time, any label y whose nonconformity score falls below τ is included in the prediction set. Full conformal is computationally expensive (requires re-fitting for each candidate label); the practical split-conformal (inductive) variant reserves a separate calibration set and requires only a single model fit, enabling deployment-time coverage guarantees on any pre-trained model. Cross-conformal and jackknife+ variants improve efficiency while retaining coverage. By 2023–2025, the theoretical community had extended conformal to Distribution Shift settings (weighted conformal prediction, covariate shift correction), sequential and online data (online conformal prediction), multi-output and structured prediction tasks, and Fairness in Machine Learning goals (equalized coverage across demographic groups). The transfer conformal setting addresses the case where calibration data comes from a source domain while test data comes from a related but different target domain, requiring domain adaptation of the nonconformity threshold.

The transition from academic curiosity to industrial practice accelerated sharply after 2022. The COPA (Conformal Prediction and its Applications) symposium grew from 22 papers in 2022 to 40 papers in 2023 (Limassol) and maintained strong participation through 2024 (Milan). TorchCP, a Python library providing conformal wrappers for PyTorch models, was released in 2024. Applications in Drug Discovery have demonstrated that conformal prediction sets over molecular property predictions reliably contain the true binding affinity with the stated probability, enabling chemists to prioritise synthesis queues under risk-controlled exploration. In Medical Imaging, conformal risk control frameworks (Angelopoulos and Bates, 2022) provide pixel-level segmentation sets with bounded false-negative rates. In Natural Language Processing and Large Language Models, conformal abstention policies trained on calibration data allow models to decline answering when uncertainty is too high, reducing hallucination rates with provable risk bounds. In Autonomous Systems and robotics, conformal prediction intervals over trajectory predictions have been integrated with control-theoretic safety filters to provide end-to-end probabilistic safety guarantees.

Mathematical Framework

The formal machinery of conformal prediction rests on four core objects: (1) a training dataset Z_train = {(x₁, y₁), …, (x_n, y_n)}; (2) a calibration dataset Z_cal = {(x_{n+1}, y_{n+1}), …, (x_{n+m}, y_{n+m})}; (3) a nonconformity score function A(Z_train, (x, y)) → ℝ producing a real number measuring how poorly (x, y) conforms to the pattern learned from Z_train; and (4) a test point (x_{n+m+1}, y_{n+m+1}) whose label is unknown.

The split conformal algorithm proceeds as follows: (a) fit a model f on Z_train; (b) for each calibration point (x_i, y_i), compute s_i = A(f, (x_i, y_i)); (c) compute the empirical quantile τ = ⌈(1 − α)(m + 1)⌉/m-th quantile of {s_{n+1}, …, s_{n+m}}; (d) output the prediction set C(x_test) = {y : A(f, (x_test, y)) ≤ τ}. The coverage guarantee is: P(y_test ∈ C(x_test)) ≥ 1 − α, where the probability is over the joint draw of calibration data and test point from the exchangeable distribution.

For Regression, C(x) is an interval [ŷ(x) − τ, ŷ(x) + τ] when using the absolute residual score; for Classification, C(x) is the set of classes {k : 1 − p̂(k | x) ≤ τ} (where p̂ is the softmax output). The key insight is that τ is computed from the calibration set once and applied to all test points, making deployment lightweight: only a single quantile lookup is needed at inference time. The set size (for classification) or interval width (for regression) automatically adapts to the model’s uncertainty through the nonconformity score: a well-calibrated model with low residuals on calibration data yields a small τ and thus narrow intervals; a poorly-calibrated model yields large τ and wide intervals, honestly reflecting its uncertainty.

The efficiency of the prediction set — its average size — depends critically on the nonconformity score design. Adaptive scores that incorporate input-conditional uncertainty estimates (e.g., normalised scores using a separate heteroscedastic variance model σ̂(x)) yield smaller, more informative sets without sacrificing coverage. The conformalized quantile regression (CQR) method (Romano et al., 2019) uses quantile regression residuals to construct asymmetric prediction intervals, adapting to heteroscedastic noise. Locally adaptive conformal methods condition the quantile estimate on neighbourhoods of the test point to improve conditional coverage, at the cost of stronger assumptions.

The connection to Hypothesis Testing is direct: a conformal p-value for label y is p(y) = |{i : s_i ≥ s(x_test, y)}| / (m + 1), and the prediction set C(x_test) = {y : p(y) > α} is exactly the acceptance region of a family of level-α tests. This reformulation enables conformal p-values to be combined with multiple testing corrections (Bonferroni, BH) for Anomaly Detection with false discovery rate control.

Comparison with Alternative Uncertainty Methods

Understanding conformal prediction’s place in the Uncertainty Quantification landscape requires comparing it against the primary alternatives:

Bayesian Inference and Bayesian Deep Learning: Bayesian methods compute a posterior distribution P(θ | data) over model parameters and produce predictive distributions P(y | x, data) = ∫ P(y | x, θ) P(θ | data) dθ. These distributions are semantically richer — they support coherent updating, decision-theoretic optimal actions, and full posterior propagation — but require specifying a prior P(θ) and a likelihood P(y | x, θ), and the posterior is typically intractable for deep networks. Approximate Bayesian methods (variational inference, Monte Carlo Methods Dropout, Laplace approximation) trade accuracy for tractability. Conformal prediction, by contrast, requires no probabilistic model specification and delivers a frequentist coverage guarantee rather than a posterior distribution. The two approaches are complementary: a Bayesian predictive distribution can serve as the nonconformity score in conformal prediction, inheriting Bayesian richness while acquiring frequentist coverage guarantees.

Ensemble Methods (Deep Ensembles): Deep ensembles (Lakshminarayanan et al., 2017) train multiple neural networks and aggregate their predictions. They are empirically well-calibrated on in-distribution data but provide no formal coverage guarantees; calibration can degrade under Distribution Shift. Conformal prediction with ensemble mean as the base predictor provides formal guarantees at any chosen α, without depending on ensemble diversity being sufficient for calibration. Ensemble-based nonconformity scores (e.g., ensemble disagreement) can be used within conformal to capture epistemic uncertainty.

Model Calibration (Temperature Scaling, Platt Scaling): Post-hoc calibration methods adjust the model’s output probabilities to match empirical frequencies. They improve the reliability of confidence scores on in-distribution data but provide no finite-sample guarantee — the calibration holds only asymptotically and only for the distribution on which it was calibrated. Conformal prediction provides a finite-sample guarantee for any held-out calibration set, regardless of the base model’s calibration quality. Bayesian Optimisation of calibration hyperparameters (temperature, Platt coefficients) via held-out validation improves empirical calibration but still cannot replace the formal coverage guarantee of conformal prediction.

Gaussian Process Regression: GPs provide analytical posterior predictive distributions with exact uncertainty under Gaussian noise and a chosen kernel, but scale as O(n³) in training data size, require kernel selection, and assume Gaussian likelihood. Conformal prediction wraps any model (including GPs) and scales to large datasets with O(n) calibration cost and O(1) inference cost (after threshold computation). For non-Gaussian problems, GPs require approximate inference and lose exact coverage, while conformal prediction maintains its coverage guarantee throughout.

MC Dropout (Gal and Ghahramani, 2016): Monte Carlo Dropout approximates Bayesian inference by applying dropout at test time and averaging over T forward passes. It is computationally cheap (factor of T over a single forward pass) and provides uncertainty estimates, but has no formal coverage guarantee and tends to underestimate uncertainty under covariate shift. Conformal prediction with a dropout-enabled model as the base predictor combines the speed of MC Dropout (through sampling-based nonconformity scores) with formal coverage guarantees. Variational Inference methods (mean-field, normalising flows) similarly provide distributional posteriors without formal coverage, making them complementary rather than competing with conformal methods.

Online and Sequential Methods: For streaming data, conformal prediction must be adapted to handle non-exchangeable sequences. Online conformal prediction (Gibbs and Candès, 2021) updates the threshold α_t at each time step using a proportional control rule based on the observed coverage indicator, maintaining average coverage over long sequences without requiring exchangeability. This connects conformal prediction to Online Learning algorithms with regret guarantees, providing a unified framework for uncertainty-aware sequential decision making.

The conclusion is that conformal prediction occupies a unique niche: it is the only practical method that provides finite-sample, distribution-free, model-agnostic coverage guarantees, at the cost of producing set-valued (rather than distribution-valued) outputs and relying on exchangeability rather than i.i.d. assumptions. Its adoption is growing most rapidly in settings where formal guarantees are required by regulators, customers, or safety standards, and where the overhead of specifying a full Bayesian model is unacceptable.

Formal Proof of Coverage

The core coverage theorem can be stated and proved succinctly. Let (x₁, y₁), …, (x_n, y_n), (x_{n+1}, y_{n+1}) be an exchangeable sequence of random variables, and let s_i = A(f, (x_i, y_i)) be the nonconformity scores (where f is fitted on a separate training set, independent of the calibration set in the split conformal variant). Define τ = ⌈(1 − α)(n + 1)⌉/n-th empirical quantile of {s₁, …, s_n} and the prediction set C(x_{n+1}) = {y : A(f, (x_{n+1}, y)) ≤ τ}.

Theorem (Vovk et al., 2005; Papadopoulos et al., 2002): Under exchangeability, P(y_{n+1} ∈ C(x_{n+1})) ≥ 1 − α.

Proof sketch: By exchangeability, the rank of s_{n+1} among {s₁, …, s_n, s_{n+1}} is uniformly distributed over {1, …, n+1}. Therefore P(rank(s_{n+1}) ≤ ⌈(1 − α)(n + 1)⌉) = ⌈(1 − α)(n + 1)⌉ / (n + 1) ≥ 1 − α. The event {rank(s_{n+1}) ≤ ⌈(1 − α)(n + 1)⌉} is equivalent to {s_{n+1} ≤ τ} = {y_{n+1} ∈ C(x_{n+1})} (by definition of the nonconformity score and prediction set). Hence the result. The proof is distribution-free: it requires no assumptions beyond exchangeability and holds for any continuous or discrete distribution, any model, and any finite n ≥ 1.

For split conformal with calibration set size m, the guarantee is P(y_{test} ∈ C(x_{test})) ≥ 1 − α, and the coverage is exactly 1 − α + O(1/m) — slightly conservative but approaching exact coverage as m grows. The exact coverage of full conformal holds with equality rather than inequality for continuous distributions.

Efficiency and Conditional Coverage

While marginal coverage is guaranteed, practitioners and regulators often care about conditional coverage — that the coverage holds for specific input values x, not just on average. Formally: P(y ∈ C(x) | X = x) ≥ 1 − α for every x. This is a strictly stronger requirement than marginal coverage.

A fundamental negative result (Venn-Abers; Foygel Barber et al., 2019) shows that achieving exact conditional coverage is generally impossible without additional assumptions: any prediction set achieving exact conditional coverage must either have infinite expected size or use knowledge of the conditional distribution P(y | x). This impossibility motivates two research directions: (a) mondrian conformal prediction, which achieves conditional coverage within pre-defined groups (classes, demographic strata) at the cost of requiring group membership labels; and (b) local and kernel-based methods that achieve approximate conditional coverage by adapting the calibration quantile to the neighbourhood of each test point.

A separate efficiency metric — the average prediction set size for classification or average interval width for regression — measures the information content of the prediction sets. Smaller sets (with the same coverage level) are more informative. Efficiency depends on the quality of the nonconformity score: a score that closely reflects the model’s true prediction uncertainty produces tightly adaptive sets. The RAPS (Regularised Adaptive Prediction Sets) score (Angelopoulos et al., 2021) is explicitly designed to minimise set size while maintaining coverage by penalising the inclusion of softmax-unlikely labels with a size penalty term.

Components and Variants

• Full Conformal Prediction — computes nonconformity scores across all training data; guarantees exact marginal coverage; computationally expensive (O(n) model re-fits for regression).

• Split (Inductive) Conformal Prediction — divides data into proper training set and calibration set; requires only a single model fit; marginal coverage guarantee holds; slightly conservative due to data splitting.

• Cross-Conformal Prediction — uses cross-validation folds to obtain nonconformity scores; improves data efficiency over split conformal; validity holds under mild conditions.

• Jackknife+ (Barber et al., 2021) — leave-one-out residuals with a corrected quantile; provides nearly exact marginal coverage with better conditional coverage than split conformal.

• Mondrian Conformal Prediction — conditions the nonconformity quantile on a taxonomic label (e.g., class), achieving class-conditional coverage for Classification problems.

• Adaptive Conformal Prediction — adjusts the nonconformity threshold at each time step to handle online and Distribution Shift settings (Gibbs and Candès, 2021).

• Conformal Risk Control (CRC) — generalises coverage to arbitrary bounded loss functions (Angelopoulos and Bates, 2022); enables risk-controlled prediction for structured outputs.

• Regularised Adaptive Prediction Sets (RAPS) — penalises large prediction set sizes to encourage efficiency while maintaining coverage (Angelopoulos et al., 2021).

• Venn Predictors — a generalisation of conformal that provides calibrated multi-probabilistic predictions (Vovk et al., 2022).

• Conformal P-values — reformulate conformal as a hypothesis testing framework, enabling false discovery rate (FDR) control for anomaly detection (Bates et al., 2023).

• Weighted Conformal Prediction — incorporates importance weights to achieve coverage under Distribution Shift between calibration and test distributions (Tibshirani et al., 2019).

  Nonconformity Score Design

  The choice of Nonconformity Score is the primary design degree of freedom and drives both coverage efficiency (set size) and conditional coverage quality:

• Regression: absolute residual |y − ŷ|; normalized residual |y − ŷ| / σ̂(x) (using a heteroscedastic variance model); quantile regression residuals for asymmetric intervals.

• Classification: 1 − p̂(y | x) (softmax score); rank-based scores (RAPS); energy-based scores; graph-structure scores for multilabel tasks.

• Structured prediction (NLP, vision): token-level entropy (TECP for LLMs, 2025); maximum softmax probability over generated tokens; ROUGE-based nonconformity for summarisation.

• Anomaly detection: isolation forest score, one-class SVM score, reconstruction error of autoencoder wrapped by conformal p-values.

  Use Cases

• Medical AI: risk-controlled image segmentation sets for pathology and radiology ensuring bounded false-negative rates; conformal abstention in diagnostic systems to flag uncertain cases for human review; conformal p-values for FDR-controlled medical entity extraction from EHR text (AAAI 2025).

• Drug Discovery: calibrated prediction sets over ADMET (absorption, distribution, metabolism, excretion, toxicity) properties; conformal uncertainty in molecular docking scores; risk-controlled hit lists for high-throughput screening. Studies published in 2024 show AstraZeneca and GSK research groups adopting conformal wrappers around graph neural network property predictors.

• Natural Language Processing and LLMs: conformal abstention (declining to answer when uncertain) with provable hallucination-rate control; token-entropy conformal prediction (TECP) for LLMs (2025); conformal verification of LLM information extraction from clinical texts.

• Computer Vision: bounding box uncertainty with conformal intervals for object detection; conformal prediction sets for image classification in safety-critical inspection; adaptive bounding box uncertainty via two-step conformal prediction (2024).

• Autonomous Vehicles and Robotics: conformal prediction over trajectory forecasts to bound collision probability; ensembles of safety filters validated by conformal methods (2025); robust control of linear systems via conformal prediction (2024).

• Time Series Forecasting: conformal prediction intervals for energy demand, financial returns, and climate variables with change-point aware calibration (2025).

• High-Energy Physics: calls for conformal calibration as a standard pipeline component in collider physics to enable reliable statistical decisions (2025).

• Process Monitoring: distribution-free process monitoring in manufacturing using conformal prediction (2025), with applications in semiconductor quality control and industrial recycling.

• Fairness: counterfactually fair conformal prediction (2025) ensuring demographic parity in coverage guarantees.

  Conformal Prediction and Active Learning

  Conformal prediction and Active Learning are deeply complementary frameworks for efficient learning under uncertainty. Active learning selects which data points to label in order to maximally reduce model uncertainty, typically guided by a query strategy such as uncertainty sampling, query by committee, or Bayesian optimisation of an acquisition function. Conformal prediction provides a principled measure of uncertainty — the size of the prediction set C(x) — that can directly serve as an active learning query criterion: examples with large conformal prediction sets are those where the model is most uncertain, making them the most informative candidates for labelling.

  This connection has been formalised in conformal active learning, where the conformal set size replaces ad hoc uncertainty metrics (entropy, margin, confidence) with a measure that carries formal coverage meaning. The advantage is that the uncertainty metric is interpretable: a prediction set containing 5 out of 10 classes means the model cannot rule out any of those 5 labels, which is directly informative to a domain expert deciding whether to query a label. Gaussian Process-based active learning (via Bayesian optimisation) naturally incorporates uncertainty through the posterior variance, but its scalability limits and kernel assumptions make conformal-based active learning attractive as a model-agnostic alternative.

  In Drug Discovery, conformal active learning has been applied to molecular property prediction: the conformal prediction set over ADMET properties guides the selection of molecules to synthesise and test experimentally, prioritising those with the most uncertain property predictions. In Medical Imaging, conformal-guided active learning for histopathology image annotation selects the most ambiguous patches for expert review. In Reinforcement Learning, conformal prediction over reward or transition models guides exploration towards high-uncertainty state-action regions.

  Conformal Prediction in Safety-Critical AI Deployment

  The intersection of conformal prediction with AI Safety and regulatory compliance is one of the most practically important dimensions of the framework. Safety-critical AI systems — autonomous vehicles, medical diagnostic AI, financial risk models — face a common challenge: the need to provide reliable, auditable uncertainty bounds that can be scrutinised by regulators, litigators, and certification bodies. Standard model calibration techniques (temperature scaling, isotonic regression) provide empirically calibrated confidences but no formal guarantees; their calibration can degrade silently under Distribution Shift, with no mechanism to detect or correct this degradation.

  Conformal prediction addresses this deployment challenge in several ways:

• Finite-sample guarantees under auditable assumptions: The guarantee P(y ∈ C(x)) ≥ 1 − α is a mathematical theorem, not an empirical approximation. Regulators can verify that the calibration set was held out properly, that the nonconformity score was computed correctly, and that the threshold was computed from the stated quantile. The proof is short and self-contained; it requires no specialist statistical knowledge to audit.

• Sensitivity to distribution shift: When the test distribution shifts relative to the calibration distribution, the exchangeability assumption is violated and conformal coverage may degrade. Weighted conformal prediction (Tibshirani et al., 2019) corrects for covariate shift by reweighting calibration scores by importance weights w(x_i) = p_test(x_i) / p_cal(x_i), restoring coverage under mild covariate shift. Online conformal methods (Gibbs and Candès, 2021) detect and correct for distribution shift in real time by adapting α based on observed coverage rates on a rolling window. Both mechanisms provide deployment monitoring capabilities that pure calibration techniques lack.

• Conservative abstention: When the prediction set C(x) is large (contains most classes, or spans a wide interval), conformal prediction naturally identifies uncertain predictions. In safety-critical systems, a threshold can be placed on the set size: if |C(x)| > k, abstain from acting and escalate to a human or fallback controller. This Selective Prediction mechanism carries the formal guarantee that on the subset of examples where the model does act, the coverage rate is at least 1 − α.

• Regulatory alignment: The EU AI Act (August 2024) requires high-risk AI systems to have “appropriate levels of accuracy, robustness and cybersecurity” documented and demonstrable. Conformal prediction provides a concrete, auditable mechanism for demonstrating accuracy bounds. The FDA’s Good Machine Learning Practice framework for AI-enabled medical devices similarly calls for calibrated uncertainty quantification. In the UK, the AI Safety Institute’s evaluation framework for foundation models includes uncertainty characterisation as a core safety property. These regulatory pressures have transformed conformal prediction from a niche academic tool into a compliance technology.

  The practical deployment workflow for a conformal-wrapped production ML system involves: (a) training the base model on Z_train; (b) computing calibration scores on a held-out Z_cal representative of the intended deployment distribution; (c) computing and storing the threshold τ as a deployment artefact; (d) at inference time, computing the nonconformity score for each test point and comparing to τ; (e) monitoring rolling coverage rates in production and triggering recalibration if coverage drops below 1 − α − ε for a tolerance ε; (f) logging all prediction sets for auditability. This workflow integrates naturally with modern MLOps platforms (MLflow, Vertex AI, SageMaker) and is increasingly supported by dedicated conformal prediction libraries.

  Academic Context

  Conformal prediction originated from algorithmic learning theory. Vladimir Vovk (Royal Holloway, University of London) and Alexander Gammerman established the foundations in the late 1990s through the transductive confidence machine (TCM). Glenn Shafer (Rutgers) contributed the martingale-based betting interpretation, connecting conformal to Information Theory and test martingales. The 2005 monograph Algorithmic Learning in a Random World (Vovk, Gammerman, Shafer) remains the primary theoretical reference. The transductive confidence machine (TCM) was the first formalisation of conformal prediction for classification; it used a kernel-based nonconformity measure and was applied to multi-label classification problems in the biomedical domain as early as 2001. The connection to Statistical Learning theory was established through Vovk’s prior work on online learning algorithms with rigorous mistake bounds, which share the algorithmic flavour of conformal prediction but operate in a different (non-probabilistic) framework.

  Ryan Tibshirani (UC Berkeley, then CMU) and Rina Foygel Barber (University of Chicago) developed the jackknife+ and cross-conformal variants, significantly improving conditional coverage in regression settings. Their jackknife+ paper (Annals of Statistics, 2021) was a landmark: it showed that a simple modification of leave-one-out cross-validation produces prediction intervals with near-exact marginal coverage, dramatically improving over the split conformal approach in finite samples. The Stanford/Berkeley group (Anastasios Angelopoulos, Stephen Bates, Emmanuel Candès, Lihua Lei, Yaniv Romano) produced a series of high-impact papers (2019–2024) on weighted conformal prediction, conformal risk control, conformal p-values, and the gentle introduction tutorial that brought conformal prediction to a wide ML audience. Their 2023 Foundations and Trends survey (Angelopoulos and Bates, 494 pages) is the standard introductory reference and has been cited thousands of times since publication.

  Key venues include NeurIPS, ICML, ICLR, AISTATS, and the dedicated COPA symposium (annual since 2012, co-located with ECML-PKDD). The COPA symposium grew from a small workshop of a dozen papers in its early years to a mature symposium with 40 submissions in 2023 and substantial industrial participation by 2024. The Journal of Machine Learning Research, Annals of Statistics, Bernoulli, and Electronic Journal of Statistics publish theory papers. The Journal of the American Statistical Association and Biometrika publish applied conformal methods for statistics-adjacent audiences. Active research groups include: Oxford (OxCSML group, uncertainty estimation; Philip Torr’s group, conformal prediction for vision); Cambridge (ML group, distribution-free inference; Carl Rasmussen and collaborators on Gaussian processes and conformal); Imperial College London (statistical machine learning; Mattias Bhatt and colleagues); UCL (Gatsby Unit, probabilistic ML; Yee Whye Teh group, Arthur Gretton’s kernel methods group); University of Edinburgh (ICSA group, Amos Storkey, uncertainty in safety-critical systems); Warwick (statistics and ML interface, Gareth Roberts group on MCMC and inference); and Royal Holloway (Vovk’s group, continuing theoretical development of Venn predictors and online conformal).

  Current Landscape (2026)

  By mid-2026, conformal prediction has transitioned from a niche academic technique into a deployment-grade uncertainty tool. Key developments since 2023:

• Regulatory drivers: The EU AI Act (effective August 2024, enforcement ramping through 2026) requires high-risk AI systems in medical, critical infrastructure, and education domains to provide calibrated uncertainty metrics. Conformal prediction satisfies these requirements with formal guarantees that regulators can audit. FDA’s Good Machine Learning Practice guidelines similarly call for uncertainty characterisation. This regulatory tailwind has driven corporate adoption, particularly in medical AI companies.

• Tooling maturity: TorchCP (2024, open-source) provides plug-in conformal wrappers for any PyTorch model. ConformalCI (2024) is a scikit-learn-compatible library. Vendor ML platforms (AWS SageMaker Clarify, Azure Machine Learning, Databricks MLflow) have begun integrating conformal calibration steps into their model deployment pipelines.

• LLM integration: The TECP (Token-Entropy Conformal Prediction) paper (2025) demonstrated that conformal abstention policies trained on LLM calibration sets can reduce hallucination rates with provable risk bounds, without requiring model retraining. COIN (2025) provides uncertainty-guarding selective question answering for foundation models with provable risk guarantees. These results are being productionised in enterprise LLM deployments.

• Scale: The awesome-conformal-prediction GitHub repository (Valeriy Manokhin) serves as a community hub and documented explosive growth in contributions from 2022 to 2025 — from dozens to hundreds of papers and implementations.

• Cross-disciplinary adoption: Physics (high-energy physics calibration standard, 2025), process engineering (semiconductor and industrial quality control), and finance (risk-controlled trading strategy selection) have published domain-specific conformal frameworks.

• Theoretical frontiers: Conditional coverage (coverage conditional on specific input x) remains an open problem — it is provably impossible without additional assumptions. Research in 2024–2025 focuses on approximate conditional coverage, local conformal methods, and conformal prediction under covariate shift and concept drift.

  UK Context

  The UK has significant conformal prediction research activity centred at institutions with strong probabilistic ML traditions. Royal Holloway, University of London is the historic home of the framework — Vladimir Vovk’s group continues theoretical development there, including Venn predictors and online conformal methods. Vovk joined Royal Holloway in the late 1990s after collaborating with Gammerman (who has been at Royal Holloway since the 1980s) and has supervised multiple PhD students and postdocs who have gone on to lead conformal prediction research groups internationally. The group’s contribution includes the theoretical foundations (algorithmic learning theory, martingale-based test statistics), Venn prediction (a multi-probabilistic extension), and online conformal prediction for sequential data.

  The Alan Turing Institute (London, headquartered at the British Library) hosts cross-disciplinary conformal prediction workshops bridging statistics and ML, and its data-centric AI programme has commissioned reports on uncertainty-aware AI deployment. Turing Fellows from Oxford, Cambridge, Edinburgh, Imperial, UCL, and Warwick contribute to the Institute’s probabilistic ML and uncertainty quantification agenda. The Turing’s Safe and Ethical AI programme explicitly addresses uncertainty quantification as a prerequisite for responsible AI deployment.

  Imperial College London’s Department of Statistics and Data Science and its Machine Learning group publish on distribution-free inference and Uncertainty Quantification. The Statistics department at Imperial maintains strong collaborations with the biomedical and environmental sciences, where conformal prediction is being applied to clinical trial design and environmental monitoring. UCL’s Gatsby Computational Neuroscience Unit (Yee Whye Teh’s group and collaborators, Arthur Gretton’s kernel methods group) works on probabilistic deep learning, kernel-based uncertainty estimation, and connections to conformal methods — UCL Gatsby is one of the world’s premier probabilistic ML research centres.

  University of Edinburgh’s ICSA group (Amos Storkey and colleagues) contributes work on uncertainty in safety-critical systems. Edinburgh’s robotics group works on conformal prediction for robot trajectory planning. Cambridge’s Computational and Biological Learning (CBL) group (Zoubin Ghahramani, Carl Rasmussen, and colleagues) work on Bayesian Inference, Gaussian processes, and distribution-free inference — Ghahramani’s group has published on the relationship between Bayesian and frequentist uncertainty methods. Cambridge’s Statistical Laboratory (Richard Samworth, Rajen Shah, and colleagues) publishes on distribution-free inference and high-dimensional statistics.

  Oxford’s OxCSML (Oxford Computational Statistics and Machine Learning) group (Dino Sejdinovic, Yarin Gal, and others) publishes on uncertainty estimation, kernel methods, and Bayesian Deep Learning. Yarin Gal’s PhD thesis on uncertainty in deep learning (2016, Cambridge) introduced MC Dropout and became one of the most cited papers on uncertainty in neural networks; Gal’s subsequent Oxford group has continued publishing on Bayesian approximations and their relationship to conformal methods.

  Warwick’s Department of Statistics has a long tradition of mathematical statistics research relevant to conformal prediction, including work on extreme value theory, multiple testing, and sequential inference. The Warwick Statistics group has collaborated with the Alan Turing Institute on uncertainty quantification for scientific AI applications.

  In Northern England, Sheffield’s Machine Learning group (founded by Neil Lawrence, now Professor at Cambridge) has been influential in probabilistic ML and Gaussian processes; the group continues this tradition under successors including Mauricio Álvarez and collaborators. Leeds and Manchester contribute through their statistics departments: Leeds’s Statistics group works on survival analysis and time-to-event prediction, where conformal prediction sets provide valid prediction intervals for survival times. Manchester’s School of Mathematics (Statistics and Probability group) publishes on distribution theory and has applied conformal methods to financial risk modelling.

  Industrially, UK-based pharmaceutical companies AstraZeneca (Cambridge) and GSK (London/Stevenage) have published work on conformal prediction for Drug Discovery ADMET property prediction. AstraZeneca’s Molecular AI group is particularly active, having published on conformal prediction wrappers for graph neural network property predictors in 2023–2024. GSK’s AI/ML team has adopted conformal calibration for early drug discovery screening. The UK’s AI Safety Institute (established 2023, housed at DSIT) focuses on frontier AI risk including uncertainty quantification as part of safety evaluations, and has cited conformal prediction as a relevant technique in its published evaluation methodology for large language models. Wayve (autonomous vehicles, London) has published work on uncertainty-aware planning relevant to conformal prediction for trajectory forecasting.

  Future Directions (2026–2030)

• Conditional coverage solutions: Current research on kernel-based local conformal methods, neighbourhood-adaptive quantile estimation, and distribution-free conditional tolerance regions aims to close the gap between marginal and conditional guarantees. The theoretical impossibility of exact conditional coverage motivates research on approximately conditional coverage with quantified approximation error, and on learning auxiliary models that predict conditional coverage rates. By 2028, a family of practical locally-adaptive conformal methods with provable approximate conditional coverage guarantees and efficient computation is expected.

• Foundation model integration: Conformal wrappers will be natively integrated into major LLM APIs (OpenAI, Anthropic Claude, Google Gemini) as a reliability layer, producing calibrated abstention signals alongside generated text. Token-level conformal uncertainty (TECP and descendants) will become standard components of production LLM serving stacks, enabling per-request confidence scoring with formal risk bounds. This has applications in enterprise deployments where liability for incorrect AI outputs requires demonstrable uncertainty bounds.

• Multi-agent and agentic AI safety: Conformal prediction propagated through multi-step agentic reasoning chains to maintain end-to-end coverage in Reinforcement Learning and task execution systems. When an AI agent takes a sequence of actions based on uncertain predictions, the per-step conformal guarantees must be composed to yield an end-to-end coverage guarantee for the full action sequence. This requires careful accounting of how calibration errors compound across steps, an open theoretical problem as of 2026.

• Federated conformal prediction: Calibration without sharing raw data — privacy-preserving conformal prediction for hospital networks (work published in 2025 using conformal prediction for privacy-preserving ML, arXiv:2507.09678). Federated calibration distributes the nonconformity score computation across data-owning nodes, aggregating only the quantile threshold without sharing individual calibration examples. Differential privacy mechanisms applied to the quantile computation provide formal privacy guarantees alongside coverage guarantees.

• Causal conformal prediction: Integration with causal inference frameworks to provide coverage guarantees under interventional distribution shift. When a Distribution Shift arises from an intervention (a policy change, a treatment, a deployment context change) rather than from natural covariate shift, standard weighted conformal prediction may fail because the importance weights are not estimable from observational data. Causal conformal prediction uses causal graph structure to bound coverage under specific intervention types.

• Standardisation: ISO/IEEE standards bodies are expected to formalise uncertainty quantification requirements for AI safety by 2028, with conformal prediction as a reference method alongside Bayesian credible intervals and bootstrap confidence intervals. The IEEE P2940 standard (AI system safety) and ISO/IEC 42001 (AI management systems) are the most likely vehicles for this standardisation. Conformal prediction’s finite-sample, distribution-free guarantee makes it the natural reference method for regulatory standards that must apply across diverse AI system types.

• Hardware acceleration: GPU-accelerated conformal calibration routines enabling millisecond-latency prediction set construction for real-time applications (autonomous vehicles, trading systems, real-time medical monitoring). The calibration computation (sorting, quantile extraction) is O(n log n) in the calibration set size and trivially parallelisable; hardware-optimised implementations are expected to enable conformal prediction on edge devices with sub-millisecond overhead.

• Conformal prediction for scientific AI: Integration into scientific computing workflows for climate modelling, protein structure prediction, and materials discovery. GraphCast and related atmospheric AI models will incorporate conformal uncertainty bands over gridded forecast fields; AlphaFold-type protein structure predictors will provide conformal confidence regions over atomic coordinates; materials discovery platforms will provide conformal prediction sets over synthesis outcomes, enabling formal risk control in autonomous laboratory systems.

  Conformal Prediction for Structured and Multi-Output Predictions

  The extension of conformal prediction beyond scalar Regression and single-label Classification to structured and multi-output prediction tasks represents a major research frontier. These settings include: Multi-Label Classification (predicting a set of relevant labels); Object Detection (predicting bounding boxes, classes, and confidences); Image Segmentation (predicting pixel-level class assignments); natural language generation (predicting a sequence of tokens); and Time Series Forecasting (predicting a vector or distribution over future time steps).

  The challenge in structured prediction is that the nonconformity score must be defined over complex objects (sequences, sets, images), and the prediction set must be a collection of such objects. Two approaches have been developed:

  Risk-based conformal prediction (Conformal Risk Control, Angelopoulos and Bates, 2022) generalises conformal to arbitrary bounded loss functions L(ŷ, y) ∈ [0, 1]. Rather than requiring y ∈ C(x), it requires E[L(ŷ(x), y)] ≤ α. This covers: image classification recall (fraction of true labels included in the prediction set); object detection recall (fraction of ground-truth boxes covered by prediction boxes with sufficient IoU); and summarisation quality (ROUGE score ≥ target). The algorithm calibrates a threshold on the loss function over the calibration set and applies it at test time, with the same finite-sample guarantee structure as standard conformal.

  Nested prediction sets provide conformal coverage for sequence prediction by treating the entire sequence as a unit. For Large Language Models, conformal prediction over token sequences defines a prediction set as a collection of complete token sequences; the nonconformity score compares the sequence-level loss (e.g., negative log-likelihood) of the true continuation against the calibration distribution. The prediction set at coverage level 1 − α contains all sequences whose log-likelihood exceeds the α-quantile threshold, which corresponds to beam search with a formal coverage guarantee.

  Functional prediction sets for Time Series Forecasting produce bands over future trajectories rather than scalar intervals. Conformal prediction for time series must account for temporal dependence (violating the i.i.d./exchangeability assumption); adaptive conformal inference (Gibbs and Candès, 2021) and online conformal prediction methods (Zaffran et al., 2022) maintain coverage in non-stationary time series by adapting the threshold in response to observed coverage violations. The 2025 paper on conformal prediction for time series with change points (arXiv:2509.02844) develops specific methods for detecting and adapting to structural breaks in the time series, maintaining coverage through distributional regime changes.

  Multi-output and graph-level predictions for molecular property prediction (relevant to Drug Discovery) predict vectors of ADMET properties simultaneously. Multi-output conformal prediction extends coverage to joint prediction sets, requiring all properties to lie within their respective intervals with joint probability 1 − α. Bonferroni correction provides a simple but conservative approach; more efficient methods use the max-score nonconformity measure or ellipsoidal prediction sets calibrated on the calibration set’s multivariate residual distribution.

  Conformal Prediction in Natural Language Processing

  The application of conformal prediction to Natural Language Processing and Large Language Models has rapidly become one of the most active research fronts, driven by the high stakes of LLM deployment in enterprise and regulated contexts. Standard large language models produce token-level probability estimates, but these are poorly calibrated — models with extremely high stated confidence often produce hallucinations, and the probability of an entire generated sequence is dominated by token-level uncertainties that do not cleanly map to semantic correctness. Semantic Entropy (Farquhar et al., 2024, Nature) provides a complementary technique generating multiple samples and clustering them by semantic equivalence via an entailment model, though conformal methods offer the advantage of formal coverage proofs that semantic entropy does not.

  Conformal Language Modeling (Quach et al., 2023, ICLR) was a pioneering work treating LLM generation as a sampling problem: the algorithm samples candidate completions from the LLM until a calibrated stopping rule (based on conformal calibration of a quality function over the calibration set) guarantees that the output set contains at least one acceptable completion with high probability. The quality function can be any black-box scorer (ROUGE, BERTScore, a human preference model), enabling task-specific conformal guarantees for diverse generation tasks. The framework connects naturally to Retrieval Augmented Generation pipelines, where conformal calibration of retrieval scores can bound the probability that all relevant documents are retrieved in the context window.

  TECP (Token-Entropy Conformal Prediction, 2025, arXiv:2509.00461) operates at the token level, using the per-token entropy of the LLM’s probability distribution over the vocabulary as the nonconformity score. A token-level prediction set is produced at each position in the generated sequence; when the entropy-based set size exceeds a threshold (calibrated on the calibration set), the model abstains or hedges. This provides a mechanism for detecting and flagging uncertain positions in generated text, enabling downstream systems to flag high-uncertainty tokens for human review.

  COIN (2025, arXiv:2506.20178) provides uncertainty-guarding selective question answering, with the model abstaining when the conformal prediction set over candidate answers is too large, and providing a formal risk bound on the fraction of answered questions that will be incorrect. This directly addresses the LLM hallucination problem with a formal rather than empirical guarantee.

  Conformal prediction for LLM information extraction (AAAI 2025) applies conformal p-values to entity extraction from electronic health records: the model extracts medical entities (conditions, medications, procedures) and assigns conformal p-values to each extraction; only extractions with p-values above the α threshold (i.e., that conform to the calibration distribution of correct extractions) are included in the output, with FDR control over the extraction task.

  The key technical challenge in applying conformal prediction to LLMs is the absence of a natural held-out calibration set at deployment time. Conformal calibration requires data from the deployment distribution; for general-purpose LLMs deployed across diverse user queries, this calibration data may be unavailable or rapidly drifting. Solutions include: dataset-specific calibration for each deployment context (feasible for enterprise LLM deployments with known task distributions); online conformal adaptation (adjusting the threshold in real time based on user feedback); and transfer conformal methods (calibrating on a source distribution and applying correction terms for target distribution shift).

  Key Terminology

• Exchangeability: A sequence of random variables is exchangeable if its joint distribution is invariant under any finite permutation; weaker than i.i.d. and the minimal assumption conformal requires.

• Nonconformity score: A real-valued function measuring how unlike a new example is relative to the training distribution; choices include absolute residual, softmax score, and energy score.

• Marginal coverage: P(Y ∈ C(X)) ≥ 1 − α for the prediction set C; averaged over the joint distribution of (X, Y). The guarantee conformal delivers.

• Conditional coverage: P(Y ∈ C(X) | X = x) ≥ 1 − α for every x; strictly stronger and in general unachievable without assumptions.

• Coverage efficiency: The average size of the prediction set; a smaller set with the same coverage level is more informative.

• Mondrian: Class-conditional conformal variant that conditions the threshold on a group label, achieving group-conditional coverage.

• Conformal risk control: Extension of conformal to arbitrary bounded loss functions, enabling multi-class recall control, object detection recall, and summarisation quality guarantees.

• Split conformal / inductive conformal: The practically dominant variant that requires a single model fit and a held-out calibration set.

• α (alpha): The user-specified error rate; the coverage level is 1 − α (e.g., α = 0.1 → 90% coverage).

• Calibration set: The held-out data used to compute the nonconformity quantile τ; must be independent of the training set and drawn from the same exchangeable distribution as the test points.

• Prediction set C(x): The output of conformal prediction at test point x; a set of labels (classification) or an interval (regression) that is guaranteed to contain the true label with probability ≥ 1 − α.

• RAPS: Regularised Adaptive Prediction Sets; a nonconformity score for Classification that penalises the inclusion of unlikely labels, producing smaller sets without sacrificing coverage.

• CQR: Conformalized Quantile Regression; wraps a quantile regression model to produce asymmetric prediction intervals that adapt to heteroscedastic noise.

• Jackknife+: A leave-one-out cross-validation-based conformal method (Barber et al., 2021) with near-exact marginal coverage and better conditional coverage than split conformal.

• Conformal p-value: For a label y and test point x, the fraction of calibration nonconformity scores that are at least as large as the nonconformity score of (x, y); the prediction set is the set of labels with conformal p-value > α.

• Epistemic uncertainty: Uncertainty due to limited training data, which reduces as more data is collected; captured by ensemble disagreement or Bayesian posterior variance; conformal prediction provides coverage bounds regardless of whether the underlying uncertainty is epistemic or aleatoric.

• Aleatoric uncertainty: Irreducible uncertainty due to inherent noise in the data-generating process; reflected in wide conformal intervals even under infinite training data.

• Conformal risk control (CRC): A generalisation of conformal prediction to arbitrary bounded loss functions, enabling risk control over structured outputs (object detection recall, image segmentation coverage, summarisation quality).

• Weighted conformal prediction: A variant that corrects for covariate shift by reweighting calibration nonconformity scores by importance weights estimated from source and target domain densities.

• Online conformal prediction: Sequential conformal method for non-stationary time series that adapts the threshold α_t at each step using a proportional control rule, maintaining average coverage without exchangeability.

  Tooling Ecosystem (2024–2026)

  A maturing software ecosystem has made conformal prediction accessible beyond specialist statisticians.

• TorchCP (arXiv:2402.12683, 2024): open-source Python library providing conformal wrappers for any PyTorch model; supports split conformal, cross-conformal, Mondrian, and conformal risk control; integrates with PyTorch Lightning and HuggingFace Transformers. As of mid-2026 it is the most widely adopted conformal implementation for Deep Learning practitioners.

• MAPIE (Model Agnostic Prediction Interval Estimator): scikit-learn-compatible Python library for split and cross-conformal Regression and Classification; developed by Quantmetry and maintained on GitHub under MIT licence; supports time-series conformal with Jackknife+ for regression. The primary conformal library for classical Machine Learning workflows.

• crepes: Lightweight Python conformal library focused on calibrated prediction sets and regression intervals; useful for teachability and prototyping.

• Puncc (Proximus UNControlled Conformal): A Python library for distribution-aware conformal prediction maintained by Proximus, supporting covariate shift correction via importance weighting; integrates with scikit-learn and XGBoost.

• ConformalCI (2024): scikit-learn-compatible coverage testing and calibration library with built-in integration for MLflow experiment tracking; designed for MLOps deployment pipelines.

• CREPES (Conformal Regressors and Predictors for Ensembles): specialised conformal library wrapping ensemble models to produce calibrated prediction intervals from random forests and gradient-boosted trees.

• Vendor integrations: AWS SageMaker Clarify (added conformal calibration step in 2025 preview), Azure Machine Learning (conformal prediction set as a pipeline component), and Databricks MLflow (conformal threshold logging in experiment tracking) are incorporating conformal steps natively into the ML deployment pipeline, signalling enterprise maturation of the technique.

• awesome-conformal-prediction (Valeriy Manokhin, GitHub): A community-maintained curated list of papers, software, and tutorials; grew from a few dozen resources in 2022 to hundreds by mid-2025, reflecting the explosion of research activity. Serves as the de facto community hub and literature index for the field.

  Interaction with MLOps and Model Deployment

  Conformal prediction integrates naturally into MLOps deployment pipelines as a post-training calibration step that is independent of the model training process. The workflow is: (1) train the base model (any architecture — Neural Network, gradient-boosted tree, Gaussian Process); (2) hold out a calibration set representative of the deployment distribution; (3) compute nonconformity scores on the calibration set and store the quantile τ as a deployment artefact alongside the model weights; (4) at inference time, compute the nonconformity score for each test query and apply τ to construct the prediction set; (5) log prediction set sizes and observed coverage rates to a monitoring dashboard for ongoing calibration health tracking.

  This deployment architecture integrates with MLOps platforms as follows: MLflow (model registry) stores τ as a model artefact alongside weights; Kubeflow Pipelines or Airflow can orchestrate the calibration step as a post-training pipeline stage; Prometheus and Grafana monitor rolling coverage rates and trigger alerts when coverage drops below 1 − α − ε; recalibration pipelines (computing a fresh τ on recent production data) run on a schedule to handle distribution drift. The integration with Observability tooling is particularly valuable: conformal prediction set sizes provide a model-agnostic uncertainty signal that is interpretable by operations teams without requiring statistical expertise.

  For Large Language Models deployed via API, conformal calibration operates at the application layer: a calibration dataset of (query, reference_answer) pairs is assembled, nonconformity scores (e.g., ROUGE between generated response and reference, or embedding distance) are computed on the LLM’s outputs for each calibration query, and the τ quantile is stored. At serving time, each generated response’s nonconformity score is compared to τ; responses above τ (too dissimilar from calibration distribution) are flagged for human review or trigger a refusal. This implements a formal selective prediction policy with provable risk bounds on the fraction of flagged responses that are actually correct.

  Research and Literature

  1. Vovk, V., Gammerman, A., and Shafer, G. (2005). Algorithmic Learning in a Random World. Springer. [ISBN 978-0387001524]
  2. Shafer, G. and Vovk, V. (2008). A Tutorial on Conformal Prediction. Journal of Machine Learning Research, 9, 371–421. https://jmlr.org/papers/v9/shafer08a.html
  3. Papadopoulos, H., Proedrou, K., Vovk, V., and Gammerman, A. (2002). Inductive Confidence Machines for Regression. Proceedings of ECML 2002, 345–356.
  4. Tibshirani, R.J., Barber, R.F., Candès, E.J., and Ramdas, A. (2019). Conformal Prediction Under Covariate Shift. NeurIPS 2019. https://arxiv.org/abs/1904.06019
  5. Barber, R.F., Candès, E.J., Ramdas, A., and Tibshirani, R.J. (2021). Predictive Inference with the Jackknife+. Annals of Statistics, 49(1), 486–507. https://doi.org/10.1214/20-AOS1965
  6. Angelopoulos, A.N., Bates, S., Jordan, M., and Malik, J. (2021). Uncertainty Sets for Image Classifiers using Conformal Prediction. ICLR 2021. https://arxiv.org/abs/2009.14193
  7. Angelopoulos, A.N., Bates, S., Malik, J., and Jordan, M.I. (2021). Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control. arXiv:2110.01052.
  8. Angelopoulos, A.N. and Bates, S. (2022). Conformal Risk Control. arXiv:2208.02814.
  9. Gibbs, I. and Candès, E.J. (2021). Adaptive Conformal Inference Under Distribution Shift. NeurIPS 2021. https://arxiv.org/abs/2106.00170
  10. Bates, S., Angelopoulos, A., Lei, L., Malik, J., and Jordan, M. (2023). Testing for Outliers with Conformal P-values. Annals of Statistics, 51(1), 149–178. https://doi.org/10.1214/22-AOS2221
  11. Angelopoulos, A.N., Barber, R.F., and Bates, S. (2024). Theoretical Foundations of Conformal Prediction. Foundations and Trends in Machine Learning. https://arxiv.org/abs/2407.11074
  12. Angelopoulos, A.N. and Bates, S. (2023). A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification. Foundations and Trends in Machine Learning, 16(4), 494–591.
  13. Vovk, V. (2022). Venn–Abers Predictors. arXiv:2101.00640.
  14. Romano, Y., Patterson, E., and Candès, E.J. (2019). Conformalized Quantile Regression. NeurIPS 2019. https://arxiv.org/abs/1905.03222
  15. Ndiaye, E. and Takeuchi, I. (2022). Root-Finding Approaches for Computing Conformal Prediction Sets. Journal of Machine Learning Research, 23, 1–40.
  16. Mondrian Conformal Prediction. Vovk, V., Petej, I., Fedorova, V. (2015). COPA 2015. https://proceedings.mlr.press/v42/vovk14.html
  17. Lei, J., G’Sell, M., Rinaldo, A., Tibshirani, R.J., and Wasserman, L. (2018). Distribution-Free Predictive Inference for Regression. Journal of the American Statistical Association, 113(523), 1094–1111.
  18. Stanton, S., Maddox, W., Gruver, N., Maffettone, P., Delaney, E., Greenside, P., and Wilson, A.G. (2022). Accelerating Bayesian Optimization for Biological Sequence Design. NeurIPS 2022.
  19. Quach, V., Fisch, A., Schuster, T., Yala, A., Sohn, J.H., Jaakkola, T., and Barzilay, R. (2023). Conformal Language Modeling. ICLR 2023. https://arxiv.org/abs/2306.10193
  20. TECP: Token-Entropy Conformal Prediction for LLMs. (2025). arXiv:2509.00461. https://arxiv.org/pdf/2509.00461
  21. COIN: Uncertainty-Guarding Selective Question Answering for Foundation Models. (2025). arXiv:2506.20178.
  22. Conformal Prediction for Privacy-Preserving Machine Learning. (2025). arXiv:2507.09678.
  23. Conformal Calibration Standard for Machine Learning in High-Energy Physics. (2025). arXiv:2512.17048.
  24. Conformal Prediction for Time-series Forecasting with Change Points. (2025). arXiv:2509.02844.
  25. Distribution-Free Process Monitoring with Conformal Prediction. (2025). arXiv:2512.23602.
  26. Fair Conformal Predictors for Applications in Medical Imaging. (2021). arXiv:2109.04392. https://arxiv.org/abs/2109.04392
  27. TorchCP: A Python Library for Conformal Prediction. (2024). arXiv:2402.12683. https://arxiv.org/abs/2402.12683
  28. Conformal Prediction: A Critical Perspective on Finite Sample Theory in Medical Applications. (2025). arXiv:2512.14727.

Provenance