---
title: Uncertainty Quantification in LLMs
url: https://www.emergentmind.com/topics/uncertainty-quantification-in-llms
type: topic
---

# Uncertainty Quantification in LLMs

Uncertainty quantification (UQ) in large language models (LLMs) refers to systematically estimating the confidence or reliability of model outputs, particularly in generative or open-set scenarios. UQ identifies when LLM predictions are likely to be erroneous, incomplete, or hallucinated, serving as a critical tool for downstream applications in risk-sensitive environments, selective deferral, and model comparison. Unlike traditional classification, uncertainty in LLMs often integrates both epistemic (model/parameter) and aleatoric (data/sampling) components, and must account for the vast and high-dimensional structure of autoregressive output distributions. This article synthesizes leading methodologies, theoretical underpinnings, and empirical findings from recent research, emphasizing both practical frameworks and open challenges.

## 1. Theoretical Foundations of Uncertainty in LLMs

Modern LLMs generate outputs via sampling from autoregressive distributions over large, discrete sequence spaces. Unlike standard neural classifiers, uncertainty estimation for LLMs must typically consider:

- **Total predictive uncertainty** as the entropy over possible output sequences, reflecting the model’s support for multiple plausible completions [2510.04439].
- **Aleatoric uncertainty** (data-wise), arising from inherent ambiguity in the input or prompt, often manifesting as high entropy even for a well-trained model [2403.02509, 2402.10189].
- **Epistemic uncertainty** (model-wise), reflecting ignorance about model parameters or the generative process, especially prominent under distribution shift, limited data, or misspecified model/hyperparameters [2403.02509, 2506.10060].

Formally, let $p(s\,|\,x)$ denote the joint probability for a full output sequence $s$ conditioned on prompt $x$. The canonical measure of total uncertainty is the Shannon entropy $H^*(x) = -\sum_{s\in S} p(s|x)\log p(s|x)$. Sampling-based and structural approximations leverage this formulation as a starting point [2510.04439].

## 2. Entropy-Based and Sampling-Driven UQ

The entropy of the LLM’s output sequence distribution is a standard metric for UQ but is intractable for large $S$; thus, it must be approximated from samples. The role of unobserved output sequences—those with nonzero model probability yet unsampled—proves crucial [2510.04439]:

- **Monte Carlo entropy estimate**:
  $$
  \hat{H}_\text{naive}(x) = -\sum_{s\in A} \hat{p}(s|x)\log \hat{p}(s|x)
  $$
  where $A$ is the set of sequences observed at least once in $M$ draws and $\hat{p}(s|x)$ is the renormalized probability over $A$.

- **Correction for unobserved probability mass**:
  $$
  p_\text{unseen} = 1 - \sum_{s\in A} p(s|x)
  $$
  Fold this into the entropy:
  $$
  \hat{H}(x) = -\sum_{s\in A} p(s|x)\log p(s|x) - p_\text{unseen}\log p_\text{unseen}
  $$
  Plug-in, Good–Turing, or Chao estimators adjust $p_\text{unseen}$ for low-sample regimes.

Experimental results on Falcon-40B and TriviaQA establish that correcting for missing mass (termed “EOS-UP”) stabilizes AUROC and improves calibration, especially for small $M$ [2510.04439]. Omitting EOS or length-normalizing degrades performance.

**Practical recommendations**: Always compute sequence probabilities with EOS, avoid length normalization unless separately scored, and incorporate missing-mass corrections for robust uncertainty even with few samples [2510.04439].

## 3. Internal Model Cues and White-Box UQ

Internal model signals, especially from transformer attention maps, enable efficient, sampling-free UQ:

- **RAUQ (Recurrent Attention-based UQ)** examines the dynamic pattern where attention to the immediate predecessor token in certain transformer heads drops when generating incorrect tokens [2505.20045]. RAUQ automatically identifies uncertainty-aware attention heads and propagates per-token confidence recurrently (via a weighted sum of token probability and recurrent/confidence from the previous step), finally aggregating via the maximum sequence-level uncertainty across all layers.

  This approach is entirely unsupervised, requiring only one forward pass and no additional labeling or sampling overhead. Empirically, RAUQ outperforms prior white- and black-box UQ baselines across QA, summarization, and translation tasks, with PRR improvements of up to +0.26 and less than 1% latency overhead [2505.20045].

- **Chain-of-Thought (CoT)-Enhanced UQ** surfaces the model’s reasoning steps. CoT-UQ weights token-level or keyword-level probabilities by importance scores along CoT extractions, then aggregates these as response-wise uncertainty [2502.17214]. This approach directly addresses the LLM’s tendency for overconfidence by probing reasoning "weak points." Main AUROC improvements are averaged at +5.9% over standard UQ and are most pronounced in logical and mathematical reasoning tasks.

- **Intra- and Cross-Layer Pattern Analysis** quantifies cross-layer agreement in the internal representations. By computing the directed layer-to-layer KL divergence of softmaxed hidden representations and flattening the resulting $L\times L$ matrices as uncertainty “signatures,” this method yields robust, transferable UQ that outperforms deep probes, especially under distribution- and quantization-shifts [2603.22299].

## 4. Semantic, Knowledge-Aware, and Geometric UQ

Semantic UQ moves beyond token-probabilities, evaluating the diversity and structure of model generations:

- **Cluster and Tensor-Based Semantic Uncertainty**. Outputs are clustered via semantic similarity (e.g. cosine similarity in embedding space) and/or entailment, then entropy is computed over clusters. The multi-dimensional MD-UQ further decomposes uncertainty by constructing pairwise semantic and knowledge-aware similarity matrices, stacking these into a tensor, and applying CP/Tucker decomposition. The sum of low-rank reconstruction errors calibrates how “structured” (low uncertainty) or “scattered” the model’s answer space is. Empirical results confirm that fusing knowledge and semantic dimensions systematically outperforms one-dimensional UQ across QA tasks [2502.16820].

- **Structural Information via Directed Semantic Graphs** (SeSE) encodes sampled responses as nodes in a directed, sparsified entailment graph. Hierarchical semantic structural entropy, minimized via optimal encoding trees, reflects the compressibility—and thus uncertainty—of the semantic space [2511.16275]. SeSE exhibits substantial AUROC and AURAC gains (up to +15%) compared to semantic entropy and kernel-based baselines, and extends to fine-grained claim-level UQ in long-form text.

- **Geometric Dispersion via Convex Hulls**. Embedding LLM outputs (via BERT or similar) and quantifying the area/volume of the convex hull (post dimensionality reduction and clustering) provides a continuous, model-agnostic UQ metric. Larger hull areas reflect greater output dispersion and uncertainty, scaling with prompt complexity and temperature [2406.19712].

## 5. UQ under Model Adaptation, Ensembles, and Distillation

Parameter-efficient fine-tuning and ensemble-based approaches offer tractable posterior approximations and new axes of uncertainty estimates:

- **LoRA Deep Ensembles**. Each ensemble member is a low-rank adapter trained from a shared pretrained base, approximating the Bayesian posterior over model parameters [2402.12264]. Ensemble-averaged predictive entropy and mutual information (entropy minus mean ensemble-member entropy) decompose total, aleatoric, and epistemic uncertainty.

  Empirical analysis on Mistral-7B, CommonsenseQA, and MMLU shows ensemble-based UQ lowers NLL and ECE relative to single-member models and surfaces retention of prior knowledge under overfitting.

- **Functional-Level UQ via Adapter-augmented MoE** (UQ4CT). By hierarchically decomposing functional space via a mixture-of-experts during fine-tuning, epistemic uncertainty is defined as the expected variance between different expert outputs, weighted by gating probabilities. Functional-level calibration is enforced throughout training, yielding >25% ECE reduction across five QA tasks and maintaining robustness to distributional shift [2410.06431].

- **Distillation of Evidential UQ**. Teacher ensembles or Bayesian prompt mixtures are distilled into single-pass student LLMs with softmax or Dirichlet (evidential) output heads. Dirichlet students directly model both mean and facial epistemic uncertainty via output concentration. Such students preserve or surpass teacher accuracy and ECE while incurring a 17× speed-up at inference [2507.18366].

- **Bayesian Prompt as Parameter** (Textual Bayes). Prompts are interpreted as textual model parameters $\theta$ with an explicit (potentially language-defined) prior. Bayesian inference is performed using Metropolis–Hastings with LLM-driven proposals. The result is full posterior predictive uncertainty over both textual parameters and responses, which can be directly applied to black-box LLM APIs [2506.10060].

## 6. Perturbation-, Calibration-, and Black-Box Techniques

Perturbation- and aggregation-based frameworks and new calibration paradigms underpin robust UQ, especially when model internals are inaccessible:

- **SPUQ (Sampling with Perturbation for UQ)** exposes epistemic uncertainty via prompt and temperature perturbations. Aggregating across diversity of model outputs using inter-sample (similarity-based) or intra-sample methods, SPUQ achieves 30–70% reduction in ECE relative to standard sampling or likelihood UQ across both closed and open API LLMs, while maintaining strong accuracy-calibration alignment [2403.02509].

- **Inv-Entropy** [2506.09684] formalizes UQ via the conditional entropy $H(X|Y)$ under a learned inverse probabilistic mapping from output samples $Y$ back to perturbation-induced input distributions $X$. This fully probabilistic, random-walk-based method, combined with genetic algorithm paraphrasing and engineered for trend monotonicity under temperature shifts, consistently achieves state-of-the-art calibration, PRR, and temperature sensitivity of uncertainty (TSU) across multiple QA benchmarks.

- **Data-Driven Conditional Dependency Learning** (TAD). Model overconfidence due to neglected token dependencies is addressed by learning a regression model that predicts the gap between conditional and unconditional token probabilities, using attention features and previous confidence. This modulates token uncertainties in a single decoding pass, improving PRR by up to 0.5 across multiple generative tasks and yielding inexpensive, interpretable UQ [2408.10692].

- **Conformal and Calibration-Based UQ**. Split conformal prediction using semantic clusters or negative log-likelihood scores provides distribution-free coverage guarantees for set-valued predictions. Empirical benchmarks across 45 model-task pairs confirm that even models with high raw accuracy can exhibit systematic over- or underconfidence, as reflected by predictive set size and calibration metrics [2411.02381, 2401.12794].

## 7. Benchmarks, Open Problems, and Future Directions

Standardized uncertainty metrics include AUROC, AURAC, PRR, ECE, Brier score, and conformal prediction set size and coverage. Calibration, robustness to model and task shift, and ability to disentangle sources of uncertainty (aleatoric vs epistemic, semantic vs knowledge) remain critical evaluation dimensions [2410.22685, 2502.17773].

Table: Representative UQ Frameworks, Methodology, and Domains

| Method             | Main Principle                             | Applicability / Highlights       |
|--------------------|--------------------------------------------|----------------------------------|
| Entropy + Missing Mass [2510.04439] | Sequence entropy corrected for unobserved sequences | General, especially effective with few samples |
| RAUQ [2505.20045]        | Recurrent attention-based; no sampling | White-box LLMs, real-time UQ     |
| SPUQ [2403.02509]         | Input perturbation + sample aggregation | Black-box or API-only LLMs       |
| SeSE [2511.16275]         | Semantic graph + hierarchical encoding | Superior performance on hallucination detection |
| LoRA Ensemble [2402.12264]| Ensemble posterior over fine-tuned adapters | Parameter-efficient fine-tuning  |
| Evidential Distillation [2507.18366]| Dirichlet student from ensemble teacher | Single-pass, OOD detection       |
| Inv-Entropy [2506.09684]  | Inverse conditional entropy via random walks | Theoretically grounded, trend-monotonic UQ |
| TAD [2408.10692]          | Regression of dependency-modulated token confidence | Fast, compositional UQ in generation |
| MD-UQ [2502.16820]        | Tensor decomposition of semantic+fact similarity | High-stakes, knowledge-rich domains |
| Benchmark [2401.12794]    | Conformal prediction + accuracy/uncertainty trade-offs | Cross-model, cross-task benchmarking |

Current directions entail extending semantic/structural UQ to long-form and open-ended generation, integrating structure-aware and attention-based UQ into production LLM APIs, adapting calibration strategies for prompt ensembles or multi-agent systems, and establishing richer benchmarks decomposing uncertainty along epistemic/aleatoric and semantic/knowledge axes.

## References

- On the Role of Unobserved Sequences on Sample-based Uncertainty Quantification for LLMs [2510.04439]
- CoT-UQ: Improving Response-wise Uncertainty Quantification in LLMs with Chain-of-Thought [2502.17214]
- Uncertainty-Aware Attention Heads: Efficient Unsupervised Uncertainty Quantification for LLMs [2505.20045]
- Textual Bayes: Quantifying Uncertainty in LLM-Based Systems [2506.10060]
- SPUQ: Perturbation-Based Uncertainty Quantification for Large Language Models [2403.02509]
- Uncertainty Quantification of Large Language Models through Multi-Dimensional Responses [2502.16820]
- Addressing Uncertainty in LLMs to Enhance Reliability in Generative AI [2411.02381]
- SeSE: A Structural Information-Guided Uncertainty Quantification Framework for Hallucination Detection in LLMs [2511.16275]
- Uncertainty quantification in fine-tuned LLMs using LoRA ensembles [2402.12264]
- Inv-Entropy: A Fully Probabilistic Framework for Uncertainty Quantification in Language Models [2506.09684]
- Semantic Uncertainty Quantification of Hallucinations in LLMs: A Quantum Tensor Network Based Method [2601.20026]
- Efficient Uncertainty in LLMs through Evidential Knowledge Distillation [2507.18366]
- Uncertainty Quantification for LLM-Based Survey Simulations [2502.17773]
- Uncertainty Quantification in Large Language Models Through Convex Hull Analysis [2406.19712]
- Unconditional Truthfulness: Learning Conditional Dependency for Uncertainty Quantification of Large Language Models [2408.10692]
- Uncertainty Quantification for In-Context Learning of Large Language Models [2402.10189]
- Benchmarking LLMs via Uncertainty Quantification [2401.12794]
- Improving Uncertainty Quantification in Large Language Models via Semantic Embeddings [2410.22685]
- Functional-level Uncertainty Quantification for Calibrated Fine-tuning on LLMs [2410.06431]
- Between the Layers Lies the Truth: Uncertainty Estimation in LLMs Using Intra-Layer Local Information Scores [2603.22299]

Source: https://www.emergentmind.com/topics/uncertainty-quantification-in-llms