Uncertainty Dimensions in LLMs

• Uncertainty in LLMs is the study of diverse, measurable factors—including input ambiguity, reasoning variance, and operational instability—that impact model reliability.
• Quantification methodologies leverage metrics like token entropy, semantic clustering, and Shapley values to provide robust uncertainty estimates.
• Calibration techniques and abstention strategies based on uncertainty metrics enhance safety and decision-making in high-stakes scenarios.

Uncertainty in LLMs comprises the collection of phenomena, statistical properties, and modeling artifacts that prevent a model, its explanations, or its evaluations from offering deterministic, trustworthy outputs. Rigorous quantification of uncertainty is critical for the safe deployment, evaluation, and interpretation of LLMs—spanning use in knowledge-rich, reasoning-intensive, and high-stakes decision scenarios. Modern research has established a multi-dimensional taxonomy of uncertainty that reflects not only classical aleatoric and epistemic divisions, but also surfaces dimensions unique to LLMs: input ambiguity, reasoning variance, decoding stochasticity, operational instability, and more. Uncertainty affects both generated content and the process of evaluating or using LLMs as automated judges, and must be managed with robust, interpretable methods for calibration, prediction set construction, and abstention-based risk mitigation.

1. Foundational Taxonomy and Sources of Uncertainty

LLMs instantiate several distinct uncertainty sources, each influencing reliability and risk (Huang et al., 20 Oct 2024, Liu et al., 20 Mar 2025, Beigi et al., 26 Oct 2024):

• Input Uncertainty: Stemming from prompt ambiguity or underspecification, leading to dispersion in possible valid outputs for a given query (Liu et al., 20 Mar 2025).
• Reasoning Path (or Process) Uncertainty: Emerges in tasks requiring multi-step inference, where variance in intermediate reasoning (e.g., chain-of-thought) amplifies response unpredictability (Guo et al., 12 May 2025).
• Prediction/Decoding Uncertainty: Due to sampling randomness and model scoring variability during autoregressive generation; commonly measured by token-wise entropy or perplexity (Tomani et al., 16 Apr 2024, Sychev et al., 3 Mar 2025).
• Parameter (Epistemic) Uncertainty: Reflects knowledge gaps or mismatches in learned representations, especially acute for out-of-domain (OOD) queries (Liu et al., 2023, Beigi et al., 26 Oct 2024).
• Surface Form/Operational Uncertainty: Introduced by lexical diversity, formatting instability, or failure modes in decoding, including generation failures and self-correction dynamics (Guo et al., 12 May 2025).

Research frameworks further decompose uncertainty into surface form (lexical/syntactic), aleatoric (data ambiguity), epistemic (knowledge gap), and operational (inference/generation) uncertainty (Guo et al., 12 May 2025), providing estimators and mutual information diagnostics for metric sensitivity to each source.

2. Quantification Methodologies and Metrics

Uncertainty estimation in LLMs leverages a range of probabilistic, geometric, and language-based techniques (Liu et al., 20 Mar 2025, Catak et al., 28 Jun 2024, Kaur et al., 4 Nov 2024, Zhu et al., 29 Jul 2025):

• Token Probability and Entropy Metrics: Scalar measures computed from log-likelihoods, per-token entropy $H_i = -\sum_k p_i(k)\log p_i(k)$, and perplexity (Tomani et al., 16 Apr 2024, Liu et al., 20 Mar 2025, Xie et al., 15 Feb 2025). These capture uncertainty attributable to model scoring dispersion.
• Semantic Entropy and Clustering: Entropy over semantic clusters of sampled outputs, where clusters are formed according to soft entailment or thresholded similarity; particularly salient in generative settings (Kaur et al., 4 Nov 2024, Makridis et al., 15 Apr 2025). Semantic entropy is defined as $SE(x) = -\sum_{c \in C} p(c|x) \log p(c|x)$.
• Shapley-Based Uncertainty: Uses a continuous correlation matrix among outputs and decomposes total uncertainty via Shapley values, accounting for all inter-output dependencies and overcoming thresholding limitations of semantic entropy (Zhu et al., 29 Jul 2025).
• Probing and Consistency Metrics: Output stability is measured by comparing predictions under input perturbation (sample probing), model stochasticity (temperature probing), or output inconsistency across diverse samplings (Tanneru et al., 2023, Huang et al., 17 Aug 2024).
• Verbalized and In-Dialogue Uncertainty: Directly prompts the LLM to output confidence scores (numerical or linguistic hedging) or counts context-specific hedges (“maybe,” “possibly”) as a measure of uncertainty (Tomani et al., 16 Apr 2024, Tao et al., 29 May 2025). Linguistic verbal uncertainty (LVU) has been shown empirically to be both more interpretable and discriminative than token-probability or numerical confidence (Tao et al., 29 May 2025).
• Geometric and Embedding-Based Approaches: Response embeddings are projected into lower-dimensional spaces (via PCA on BERT embeddings), clustered (e.g., with DBSCAN), and the convex hull areas of clusters are used to reflect dispersion and thus uncertainty (Catak et al., 28 Jun 2024).
• Tensor Decomposition of Multi-Dimensional Similarity: Joint semantic and knowledge-aware similarity matrices are combined into tensors and decomposed, yielding uncertainty from reconstruction error, which robustly captures high-dimensional response diversity (Chen et al., 24 Feb 2025).
• Confusion-based Assessment for LLM-as-a-Judge: Constructs a confusion matrix over ratings generated under biased assessment prompts and derives uncertainty from the concentration and agreement structure among these outputs (Wagner et al., 15 Oct 2024).

3. Calibration, Abstention, and Performance Implications

Uncertainty estimation underpins crucial downstream behaviors, including selective prediction, abstention, and evaluation reliability. Key findings include:

• Calibration Metrics: Expected Calibration Error (ECE) is used to quantify the deviation between predicted uncertainty/confidence and observed correctness, critical for ensuring trustworthy model deployment (Liu et al., 2023, Tao et al., 29 May 2025). Calibration and selective classification (measured by AUROC) are two orthogonal axes; a model can be well-calibrated but still fail at effective error ranking (Tao et al., 29 May 2025).
• Abstention and Selective Classification: Abstaining on outputs above a certain uncertainty threshold improves accuracy, reduces hallucination rates, and raises safety, with minimal computational overhead (Tomani et al., 16 Apr 2024). Empirical results show correctness improvements of 2–8% and up to 50% reduction in hallucinated responses for high-uncertainty abstention.
• Impact of Scale and Fine-Tuning: Model scale, instruction fine-tuning, and post-training optimization (e.g., DPO, RLHF) all influence the sharpness, calibration, and expressiveness of uncertainty estimates (Tao et al., 29 May 2025). RLHF models, for example, tend to verbalize uncertainty with greater fidelity and use hedges more discriminatively (Tomani et al., 16 Apr 2024).
• In-Context Example Selection: Active selection of in-context examples using uncertainty-driven scores (e.g., output inconsistency via Unc-TTP) yields systematic performance gains, increasing model robustness in OOD and ambiguity-prone cases (Huang et al., 17 Aug 2024).
• Realignment and Hallucination: Direct manipulation of internal “verbal uncertainty” features can reduce overconfident hallucinations by ∼30%, calibrating the model’s linguistic framing to its internal semantic uncertainty (Ji et al., 18 Mar 2025).

4. Uncertainty in Evaluation and LLM-as-a-Judge Paradigms

Automated LLM-based evaluation and ranking yield additional uncertainty dimensions:

• Stochasticity in Benchmarking: LLM evaluations are inherently stochastic, even with deterministic decoding. Proper reporting of mean ± prediction interval is recommended, with sample variance and t-based prediction intervals ensuring reproducibility (Blackwell et al., 4 Oct 2024).
• Simplex-Based Ranking and Epistemic Limits: In LLM-judge scenarios, a geometric simplex model of scoring reveals a phase transition: binary grading systems permit ranking identifiability even with imperfect judges, but as scoring granularity increases (e.g., 3+ levels), rankings become fundamentally non-identifiable without prior knowledge, due to epistemic uncertainty in judge parameters (Vossler et al., 28 May 2025).
• Confusion-Matrix Approaches: For categorical evaluation, constructing the confusion matrix from biased assessments enables a distinction between high and low uncertainty, tightly correlating with evaluation correctness; this is a black-box method robust to LLM choice or access level (Wagner et al., 15 Oct 2024).
• Incorporation in Model Training: Fine-tuning LLM evaluators with explicit uncertainty features (e.g., ConfiLM) yields improved performance on out-of-distribution (OOD) benchmarks, demonstrating the value of uncertainty-aware supervision in evaluator models (Xie et al., 15 Feb 2025).

5. Multidimensional Decomposition and Adaptive Metric/Model Selection

Recent frameworks offer systematic uncertainty source decomposition and adaptive selection:

• Source Decomposition Pipelines: Multi-stage evaluation pipelines separately estimate surface-form, aleatoric, epistemic, and operational uncertainty via paraphrasing, clarification, answering, and self-checking stages. Metrics’ mutual information with each source is analyzed for insight (Guo et al., 12 May 2025).
• Adaptive Metric and Model Selection: By constructing “uncertainty profile vectors” for tasks, metrics, and models, optimal pairings are identified via cosine similarity or geometric mean, leading to 3–5% improvements in selection accuracy over baseline approaches (Guo et al., 12 May 2025).
• Practical Guidance: The resulting diagnostic vectors make uncertainty interpretable and actionable, pointing developers toward improvements targeted at the dominant uncertainty dimension for their task.

6. Theoretical and Practical Evaluation Challenges

Current research identifies persistent challenges and open questions:

• Interpretability and Attribution: Decoupling the upstream source of uncertainty—input ambiguity, reasoning instability, or model ignorance—remains a challenge. Many scalar metrics lose this granularity, necessitating multi-perspective and multi-stage approaches (Beigi et al., 26 Oct 2024, Liu et al., 20 Mar 2025).
• Transferability and Modality Alignment: Methods that calibrate uncertainty for one kind of task or domain may not generalize; extending estimation across modalities (e.g., image-plus-text) or OOD settings is an essential research trajectory (Beigi et al., 26 Oct 2024, Liu et al., 20 Mar 2025).
• Scalability and Efficiency: Bayesian and ensemble-based estimates quickly become computationally intractable at the scale of contemporary LLMs, driving pragmatic interest in single-pass or black-box quantification methods (Liu et al., 20 Mar 2025).
• Evaluation Benchmarks: Reproducible, fair benchmarks must control for domain and reasoning differences (e.g., rebalancing bias in datasets such as MMLU-Pro) to support cross-domain generalization and fair comparison (Sychev et al., 3 Mar 2025, Blackwell et al., 4 Oct 2024).

7. Implications and Future Directions

Reliable uncertainty quantification directly impacts trust, calibration, and model deployment in safety-critical domains (Tao et al., 29 May 2025, Beigi et al., 26 Oct 2024). Recent advances—including Shapley-based metrics that capture continuous semantic relationships (Zhu et al., 29 Jul 2025), conformal prediction on dynamic semantic clusters (Kaur et al., 4 Nov 2024), and multi-dimensional tensor-based frameworks (Chen et al., 24 Feb 2025)—extend beyond classical entropy and probe-based measures, offering more robust and discriminative indicators. Holistic evaluation of both calibration and error discrimination, transparent reporting (mean ± interval), and integration with abstention and active learning, are recommended for operational robustness.

The field continues to call for: multi-axis decomposition of uncertainty; development of scalable, interpretable quantification strategies; evaluations sensitive to domain and reasoning requirements; and adaptive approaches for task- and data-aware uncertainty mitigation and model calibration. These directions are critical for trustworthy LLM application in domains where error cost and risk sensitivity are paramount.