Uncertainty Quantification for In-Context Learning of Large Language Models

Chen LingXujiang ZhaoXuchao ZhangWei ChengYanchi LiuYiyou SunMika OishiTakao OsakiKatsushi MatsudaJie Ji

article2024NAACL62 citations

Presents a Bayesian framework that decomposes predictive uncertainty in large language model in-context learning into prompt-induced aleatoric and model-induced epistemic components, enabling unsupervised diagnostic evaluation of output reliability across both white-box and black-box settings.

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Large Language Models often produce unreliable outputs or hallucinations, posing significant operational and decision-making risks in practical applications. While prompting models with a few example demonstrations (in-context learning) improves performance, existing uncertainty metrics fail to distinguish whether model confusion stems from poor demonstration examples or limitations in the model itself.

The article aims to formulate and evaluate an unsupervised framework that quantifies and separates predictive uncertainty in in-context learning into two distinct sources: data-driven variation from the demonstrations and parameter-level variation from the model configurations.

The researchers evaluated this approach using natural language understanding benchmarks across sentiment analysis, topic classification, and linguistic acceptability. They tested open-source models, primarily LLaMA-2 (7B, 13B, and 70B parameter versions) and OPT-13B, across thousands of test cases. For accessible white-box models, the framework aggregates prediction entropy using beam search decoding across varying demonstration sets, while also defining a variance-based alternative for black-box systems.

The evaluation revealed several critical findings. First, decomposing uncertainty into data-level and model-level components identified misclassified samples more accurately than standard baseline metrics, such as semantic and raw token entropy. Second, ensuring demonstrations cover every label class consistently reduced error rates and uncertainty compared to purely random selection. Third, scaling up model capacity from 7B to 70B parameters significantly reduced overall uncertainty and improved error-detection accuracy. Finally, model-level uncertainty proved to be the most reliable indicator for detecting out-of-distribution or irrelevant prompt demonstrations, achieving area-under-the-curve scores exceeding 0.93.

These findings provide immediate practical value for managing generative AI risks. Organizations deploying language models can use these decomposed metrics as automated guardrails to reject low-confidence outputs, diagnose faulty prompt designs, and identify when queries fall outside the system's operational domain. Rather than treating uncertainty as a single uninterpretable number, teams can determine whether to fix prompt examples or upgrade the underlying model.

Stakeholders should implement balanced class-sampling strategies when designing few-shot prompts and integrate model-level uncertainty monitoring into production pipelines to catch out-of-domain inputs. Before deploying this approach on open-ended generation tasks like summarization or drafting, teams should run additional pilot tests, as the current framework relies on structured classification and closed-form outputs where specific answer tokens can be isolated.

Confidence in these findings is strong for deterministic natural language classification tasks across standard model architectures. However, decision-makers should remain cautious regarding open-ended text generation, where linguistic redundancy makes identifying core predictive tokens more challenging and requires further research.

Emory University Microsoft NEC Corporation NEC Laboratories America, Inc.large language models statistical language models uncertainty quantification in-context learning large language models

arXiv: 2402.10189 lingchen0331/UQ_ICL

Paper: Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods, Eyke Hüllermeier et al. (2019). Its distinction between irreducible data uncertainty and model ignorance supplies the conceptual foundation for the source’s aleatoric–epistemic decomposition.
Paper: A survey of uncertainty in deep neural networks, Jakob Gawlikowski et al. (2021). This survey explains how deep-learning methods separate data-driven from model-driven uncertainty, preparing you to understand the source’s decomposition and estimation choices.
Paper: What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?, Alex Kendall et al. (2017). Its Bayesian treatment of aleatoric and epistemic uncertainty provides an earlier applied framework for the two uncertainty sources the source measures.

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Abstract

In-context learning has emerged as a ground-breaking ability of Large Language Models (LLMs) and revolutionized various fields by providing a few task-relevant demonstrations in the prompt. However, trustworthy issues with LLM’s response, such as hallucination, have also been actively discussed. Existing works have been devoted to quantifying the uncertainty in LLM’s response, but they often overlook the complex nature of LLMs and the uniqueness of in-context learning. In this work, we delve into the predictive uncertainty of LLMs associated with in-context learning, highlighting that such uncertainties may stem from both the provided demonstrations (aleatoric uncertainty) and ambiguities tied to the model’s configurations (epistemic uncertainty). We propose a novel formulation and corresponding estimation method to quantify both types of uncertainties. The proposed method offers an unsupervised way to understand the prediction of in-context learning in a plug-and-play fashion. Extensive experiments are conducted to demonstrate the effectiveness of the decomposition. The code and data are available at: https://github.com/lingchen0331/UQ_ICL.

Topics

large language models statistical language models

Keywords

uncertainty quantification in-context learning large language models semantic uncertainty entropy-based uncertainty estimation predictive uncertainty uncertainty decomposition epistemic uncertainty aleatoric uncertainty Stanford Sentiment Treebank

1 Introduction
2 Uncertainty Decomposition of In-context Learning
2.1 Background
2.2 Predictive Uncertainty Formulation of In-context Learning
2.3 Entropy-based Decomposition
2.4 Entropy Approximation
3 Related Works
4 Experiments
4.1 Experiment Setup
4.2 Quantitative Analysis
4.3 Generalization Capability
4.4 Misclassification Rate with Out of Domain Demonstration
4.5 Out-of-domain Demonstration Detection
4.6 Semantic Out-of-distribution Detection
5 Conclusion
Limitations
References
A Appendix
A.1 Variance-based Decomposition
A.2 Dataset Description
A.3 Experiment Setup
A.4 Prompt Template
A.5 Case Study

Knowls

Knowl 1 — Latent-concept Bayesian formulation of in-context prediction
model/method
The paper models in-context learning as prediction under two sources of variation: a latent task concept inferred from demonstrations and model parameters or configurations. The prompt $x_{1:T}$ consists of demonstrations $x_1,\ldots,x_{T-1}$ and a test input $x_T$ ; each demonstration is assumed to be drawn independently from data associated with the same latent concept $z$ . The test output is $y_T$ , and $\Theta$ denotes model parameters or decoding configurations. The proposed predictive distribution is
$p(y_T\mid x_{1:T})\approx \int\!\int p(y_T\mid \Theta,x_{1:T},z)\,p(z\mid x_{1:T})\,q(\Theta)\,dz\,d\Theta.$
Here $q(\Theta)$ is an approximate distribution over model parameters or configurations, and $p(z\mid x_{1:T})$ represents uncertainty over the concept associated with the prompt. The paper approximates the conditional output likelihood using a Gaussian centered on a deterministic generation function, with covariance representing variation associated with model parameters. Treating in-context learning as Bayesian inference over latent concepts is presented as a probabilistic hypothesis about how demonstrations guide a model, not as an established account of the mechanism.
Knowl 2 — Entropy-based epistemic and aleatoric decomposition
equation
For a fixed prompt $x_{1:T}$ and model setting $\Theta$ , the paper defines total predictive uncertainty using the entropy $H(y_T\mid x_{1:T},\Theta)$ . Averaging the conditional output entropy over latent concepts (which are approximated through different demonstration sets) gives the paper's epistemic uncertainty (EU); the difference between total entropy and that average is its aleatoric uncertainty (AU):
$\mathrm{EU}=\mathbb{E}_{z}\!\left[H(y_T\mid x_{1:T},z,\Theta)\right],\qquad \mathrm{AU}=H(y_T\mid x_{1:T},\Theta)-\mathbb{E}_{z}\!\left[H(y_T\mid x_{1:T},z,\Theta)\right].$
Equivalently, the paper identifies AU with the mutual information between the output and latent concept conditional on the model setting, $I(y_T;z\mid\Theta)$ . In this decomposition, the first term is the paper's measure of EU and the entropy difference is its measure of AU; the labels follow the paper's formulation.
Knowl 3 — Answer-focused entropy estimator for free-form model outputs
model/method
To estimate the entropy components for a task with $K$ possible answer labels, the method discards generated text that does not express the answer and uses the probabilities of answer-relevant tokens. For each of $L$ demonstration sets, it generates multiple candidate responses, using beam search as a practical approximation to sampling model outputs, and aggregates the answer-token probabilities into a $K$ -dimensional column. Let $A_{k,j}$ be the accumulated probability for label $k$ under demonstration set $j$ , so $A$ is a $K\times L$ matrix. Define $\sigma(v)_k=v_k/\sum_r v_r$ to normalize a nonnegative vector, and let $H(p)=-\sum_{k=1}^{K}p_k\log p_k$ be discrete entropy. The estimates are
$\widehat{\mathrm{EU}}=\frac{1}{L}\sum_{j=1}^{L}H\!\left(\sigma(A_{:,j})\right),\qquad \widehat{\mathrm{AU}}=H\!\left(\sigma\!\left(\sum_{j=1}^{L}A_{:,j}\right)\right)-\widehat{\mathrm{EU}}.$
Thus, EU is the average entropy of the answer distribution for individual demonstration sets, while AU is the entropy of the pooled answer distribution minus that average. The procedure is designed for structured-answer tasks such as classification: the model is instructed to return only the desired label, or answer tokens are extracted from longer responses. This entropy estimator requires access to generated-token probabilities, so it is intended for white-box models.
Knowl 4 — Variance decomposition for black-box language models
equation
When token probabilities are unavailable, the paper proposes a variance-based alternative using the law of total variance. For output random variable $y_T$ , prompt $x_{1:T}$ , and model-setting distribution $q(\Theta)$ , it writes
$\operatorname{Var}(y_T\mid x_{1:T})= \operatorname{Var}_{q(\Theta)}\!\left(\mathbb{E}[y_T\mid x_{1:T},\Theta]\right) +\mathbb{E}_{q(\Theta)}\!\left[\operatorname{Var}(y_T\mid x_{1:T},\Theta)\right].$
The variance across conditional mean outputs as $\Theta$ varies is designated epistemic uncertainty; the expected conditional output variance is designated aleatoric uncertainty. In black-box use, outputs are collected by querying the model with different demonstrations and configurations, such as temperature or top- $p$ settings, and the corresponding variances are estimated from those outputs. Unlike the entropy estimator, this decomposition does not require access to token probabilities.
Knowl 5 — Evaluation datasets and inference protocol
experimental setup
The main experiments use LLaMA-2 chat models with 7B, 13B, and 70B parameters; OPT-13B is used for a cross-backbone comparison. Test sets comprise EMOTION (2,000 examples, six classes), Financial Phrasebank (850, three classes), SST-2 (872, two classes), CoLA (1,040, two classes), and AG_News (1,160, four classes). Demonstrations are sampled either randomly or with at least one example from each class. The numbers of demonstrations for random/class sampling are: EMOTION, 6/1 per class; Financial Phrasebank, 6/2 per class; SST-2, 4/2 per class; CoLA, 2/1 per class; and AG_News, 4/1 per class. Demonstrations are sampled four times; decoding uses beam search with width 10 and a maximum of 16 new tokens.

For misclassification assessment, each test prediction is labeled correct or incorrect, and uncertainty scores are evaluated using AUPR and AUROC, with larger scores intended to identify errors. Comparisons include likelihood-based uncertainty, token entropy, semantic uncertainty, and the proposed EU and AU scores.
Knowl 6 — Uncertainty scores identify errors across tasks, with dataset-dependent results
empirical result
Across the five natural-language-understanding datasets and LLaMA-2 model sizes, the proposed EU and AU measures generally provide stronger AUPR and AUROC for distinguishing misclassified from correct predictions than likelihood, token-entropy, and semantic-uncertainty baselines, though the advantage is not uniform across every dataset, model, and metric. For example, on Financial Phrasebank with LLaMA-2-70B and class-balanced demonstrations, EU reaches AUPR 0.893 and AUROC 0.804, compared with semantic uncertainty at 0.774 and 0.649, respectively. Class-balanced demonstration sampling generally performs better than random sampling, and larger models tend to improve uncertainty-assessment performance, but the paper reports exceptions to both trends. A separate EMOTION comparison reports similar qualitative precision-recall and ROC behavior for OPT-13B and LLaMA-2-13B, supporting use across backbones.

Knowl 7 — Demonstration shifts reduce AUROC more for AU than for EU

empirical result

On EMOTION with LLaMA-2-13B, the paper evaluates how well EU and AU rank misclassified examples when demonstrations come from the task's original training data, a related sentiment dataset (Financial Phrasebank), or an unrelated acceptability dataset (CoLA). The reported AUROCs for random/class demonstration sampling are:

	Random		Class
Demonstration source	EU	AU	EU	AU
Original	0.681	0.585	0.686	0.599
Relevant	0.688 (+1.0%)	0.541 (-7.5%)	0.671 (-2.2%)	0.524 (-12.5%)
OOD	0.671 (-1.4%)	0.501 (-13.3%)	0.673 (-1.8%)	0.497 (-17.0%)

Relative changes are reported against the corresponding original-demonstration score. EU changes comparatively little, whereas AUROC for AU declines substantially with shifted demonstrations, by as much as 17.0% for class-balanced sampling with CoLA demonstrations.

Knowl 8 — EU detects out-of-domain demonstrations in the tested sentiment task

empirical result

For LLaMA-2-13B on EMOTION, the paper tests whether uncertainty scores distinguish in-domain demonstrations from relevant demonstrations drawn from Financial Phrasebank or out-of-domain (OOD) demonstrations drawn from CoLA. In-domain demonstrations are labeled 0 and shifted demonstrations 1. The reported AUPR/AUROC values are:

Demonstration comparison	Score	AUPR	AUROC
Relevant vs. in-domain	Semantic	0.702	0.644
Relevant vs. in-domain	EU	0.742	0.935
Relevant vs. in-domain	AU	0.657	0.682
OOD vs. in-domain	Semantic	0.698	0.712
OOD vs. in-domain	EU	0.784	0.941
OOD vs. in-domain	AU	0.773	0.607

EU has the highest AUPR and AUROC in both comparisons. The result indicates that, in these experiments, EU is a stronger indicator of demonstration-domain shift than AU or semantic uncertainty.

Knowl 9 — EU identifies semantic out-of-distribution test examples

empirical result

The semantic out-of-distribution (SOOD) experiment uses EMOTION, masks the sadness and anger classes from the task description, and asks the model to classify test inputs among the remaining four classes. Samples from the masked classes are labeled SOOD; other samples are in-distribution. For LLaMA-2-7B and LLaMA-2-13B, the reported AUPR/AUROC values are:

Model	Semantic		EU		AU
	AUPR	AUROC	AUPR	AUROC	AUPR	AUROC
7B	0.477	0.532	0.548	0.658	0.461	0.570
13B	0.417	0.468	0.525	0.592	0.414	0.437

EU gives the strongest AUPR and AUROC for both model sizes in this setup. The authors attribute increased uncertainty for SOOD cases to the mismatch between the examples and demonstrations and to the prompt excluding the correct class.

Knowl 10 — Scope and access limitations of the uncertainty methods
limitation
The paper's evaluation and proposed answer-focused entropy estimation target natural-language-understanding tasks with identifiable answer tokens, including classification and multiple-choice questions. For open-ended generation, the method may not reliably determine which parts of a generated sequence carry semantic importance, limiting its applicability. Entropy-based estimation also needs token probabilities and therefore applies to white-box models; the variance-based alternative is offered for black-box models but does not remove the difficulty of identifying meaningful answers in unconstrained generation.

Coverage note — The detailed prompt template and the single-query case study are omitted because they illustrate the evaluated procedure rather than contribute a separate method or generalizable result.

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Citation

MLA

Ling, C., et al. “Uncertainty Quantification for In-Context Learning of Large Language Models”. Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers), 2024, pp. 3357–70, https://doi.org/10.18653/v1/2024.naacl-long.184.

APA

Ling, C., Zhao, X., Zhang, X., Cheng, W., Liu, Y., Sun, Y., Oishi, M., Osaki, T., Matsuda, K., Ji, J., Bai, G., (赵亮), L. Z., & Chen, H. (2024). Uncertainty Quantification for In-Context Learning of Large Language Models. Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers), 3357–3370. https://doi.org/10.18653/v1/2024.naacl-long.184

Chicago

Ling, C., X. Zhao, X. Zhang, et al. 2024. “Uncertainty Quantification for In-Context Learning of Large Language Models”. Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers), 3357–70. https://doi.org/10.18653/v1/2024.naacl-long.184.

Harvard

Ling, C. et al. (2024) “Uncertainty Quantification for In-Context Learning of Large Language Models”, Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers). Association for Computational Linguistics, pp. 3357–3370. Available at: https://doi.org/10.18653/v1/2024.naacl-long.184.

Vancouver

1. Ling C, Zhao X, Zhang X, et al (2024) Uncertainty Quantification for In-Context Learning of Large Language Models. In: Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers). Association for Computational Linguistics, pp 3357–3370

BibTeX

@inproceedings{ling-etal-2024-uncertainty,
    title = "Uncertainty Quantification for In-Context Learning of Large Language Models",
    author = "Ling, Chen  and
      Zhao, Xujiang  and
      Zhang, Xuchao  and
      Cheng, Wei  and
      Liu, Yanchi  and
      Sun, Yiyou  and
      Oishi, Mika  and
      Osaki, Takao  and
      Matsuda, Katsushi  and
      Ji, Jie  and
      Bai, Guangji  and
      Zhao, Liang  and
      Chen, Haifeng",
    editor = "Duh, Kevin  and
      Gomez, Helena  and
      Bethard, Steven",
    booktitle = "Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers)",
    month = jun,
    year = "2024",
    address = "Mexico City, Mexico",
    publisher = "Association for Computational Linguistics",
    url = "https://aclanthology.org/2024.naacl-long.184/",
    doi = "10.18653/v1/2024.naacl-long.184",
    pages = "3357--3370"
}

Metadata:ACL Anthology

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Uncertainty Quantification for In-Context Learning of Large Language Models

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Latent-concept Bayesian formulation of in-context prediction

Knowl 2 — Entropy-based epistemic and aleatoric decomposition

Knowl 3 — Answer-focused entropy estimator for free-form model outputs

Knowl 4 — Variance decomposition for black-box language models

Knowl 5 — Evaluation datasets and inference protocol

Knowl 6 — Uncertainty scores identify errors across tasks, with dataset-dependent results

Knowl 7 — Demonstration shifts reduce AUROC more for AU than for EU

Knowl 8 — EU detects out-of-domain demonstrations in the tested sentiment task

Knowl 9 — EU identifies semantic out-of-distribution test examples

Knowl 10 — Scope and access limitations of the uncertainty methods

References

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