Token-Level Uncertainty Quantification

• Token-level uncertainty quantification is a framework that assesses model confidence for each token, enabling detailed detection of hallucinations.
• It decomposes uncertainty into aleatoric (entropy) and epistemic (KL divergence) components using methods like feature-gap projection and attention fusion.
• Practical applications span question answering, fact-checking, and selective generation, with improvements shown in metrics such as ROC-AUC and Prediction–Rejection Ratio.

Token-level uncertainty quantification (UQ) in LLMs is a framework for measuring and interpreting a model’s uncertainty about individual token choices during autoregressive text generation. Unlike sequence-level metrics, token-level UQ directly exposes local structure in model confidence and epistemic uncertainty, enabling fine-grained detection of hallucinations, improved calibration, and targeted interventions during decoding. Recent research has developed a variety of theoretically grounded and empirically validated approaches for extracting, interpreting, and operationalizing token-level uncertainty, with applications across question answering, fact-checking, selective generation, and dialog systems.

1. Formal Foundations and Decomposition of Token-Level Uncertainty

Let $\mathcal{V}$ denote the vocabulary of an autoregressive LLM with parameters $\theta$ generating a token sequence $(y_1, y_2, ..., y_T)$ conditioned on prompt $x$ and optional context $c$. At generation step $t$, the model emits a predictive distribution $P(y_t \mid y_{<t}, x, c, \theta)$. A foundational, theoretically principled measure of the model’s token-level uncertainty is the cross-entropy from the (unknown) true next-token distribution $P^*(y_t \mid y_{<t}, x, c)$ to the model’s prediction: $$U(t) := -\sum_{y_t\in\mathcal{V}} P^*(y_t \mid y_{<t}, x, c) \cdot \ln P(y_t \mid y_{<t}, x, c, \theta)$$ This uncertainty decomposes as

$$U(t) = H[P^*(\cdot \mid y_{<t}, x, c)] + \mathrm{KL}[P^*(\cdot \mid y_{<t}, x, c) \| P(\cdot \mid y_{<t}, x, c, \theta)]$$

where $H$ denotes (aleatoric) entropy—irreducible data-level uncertainty—and the KL divergence term quantifies model epistemic uncertainty. While $P^*$ is not directly accessible, various strategies are used to approximate or upper bound this decomposition depending on the setting and the desired specificity of uncertainty attribution (Bakman et al., 3 Oct 2025, Shorinwa et al., 2024).

2. Operational Models: Approximations and Feature-Based Interpretability

The canonical intractability of $P^*$ motivates proxy constructions:

• Idealized prompting: Approximate $P^*$ by a perfectly prompted (“ideal”) version of the same architecture, denoted $\theta^*$, yielding a practical epistemic term

$$\mathrm{Epi}(t) = \mathrm{KL}[P(\cdot \mid y_{<t}, x, c, \theta^*) \| P(\cdot \mid y_{<t}, x, c, \theta)]$$

The upper bound on this KL is given by the norm of the difference in hidden activations: $\Vert h_t^* - h_t \Vert$ (Bakman et al., 3 Oct 2025).

• Linear feature decomposition: Assuming a meaningful basis of semantic feature directions $v_i$, the hidden-state difference can be decomposed as

$$\Vert h_t^*-h_t \Vert = \Vert \sum_i (\beta_i - \alpha_i) v_i \Vert$$

Each $(\beta_i - \alpha_i)$ represents a “feature gap,” mapping epistemic uncertainty at the token level onto interpretable axes such as context reliance, comprehension, and honesty, which can be extracted using a small labeled set via contrastive prompting and singular value decomposition (SVD).

• Attention-based fusion: Attention patterns in selected “uncertainty-aware” heads show sudden drops in attention to preceding tokens during incorrect generations; recurrent aggregation of attention, token probabilities, and conditional dependence enables efficient plug-and-play real-time uncertainty scoring (Vazhentsev et al., 26 May 2025).
• Density-based metrics: Mahalanobis distance (MD) is adapted to generative settings by fitting centroids and covariances of token embeddings from correct sequences, with layerwise MD features aggregated and regressed against performance labels (Vazhentsev et al., 20 Feb 2025).

3. Core Algorithms and Practical Estimation

The table below summarizes representative token-level UQ methodologies, emphasizing calculation scope, core statistic, and computational cost:

Method	Core Statistic	Overhead
Cross-entropy decomposition	KL($P^*$
Feature-gap projection	$\Vert \sum_i (\beta_i-\alpha_i) v_i\Vert$	1 forward pass + dot products
Attention chain fusion	Recurrent combination of $P(y_t|...)$, attention	1 forward pass
Mahalanobis distance	MD in latent space per layer	1 forward pass
MC Dropout / Bayesianization	Predictive entropy, mutual information	$M$ forward passes ($M>1$)
Black-box sampling/entropy	Token entropy from $M$ samples	$M$ queries (API)

Statistical proxies such as negative log-probability, entropy, and mutual information (via ensembles or perturbations) serve as fast, model-agnostic uncertainty surrogates in white-box and black-box settings (Shorinwa et al., 2024, Bakman et al., 3 Oct 2025, Zhang et al., 16 May 2025, Xu, 30 Aug 2025). For uncertainty-aware post-training, masked MLE and self-distillation focus representational capacity on high-epistemic-uncertainty tokens while maintaining generalization (Liu et al., 15 Mar 2025).

4. Empirical Validation and Comparative Performance

Empirical studies consistently report superior token-level uncertainty discrimination and hallucination detection by incorporating internal model features and hierarchical conditioning:

• The feature-gap approach, ensembling context reliance, comprehension, and honesty features, outperforms both sampling-free and sampling-based baselines (e.g., SAPLMA, Semantic Entropy) with up to 16-point improvement in Prediction–Rejection Ratio (PRR) and minimal computational cost (Bakman et al., 3 Oct 2025).
• Attention-based fusion (RAUQ) attains token-level ROC-AUCs of 0.65–0.75 (vs 0.55–0.60 for token entropy) and <1% added latency, demonstrating per-token hallucination localization capability (Vazhentsev et al., 26 May 2025).
• Mahalanobis distance regression methods provide state-of-the-art out-of-domain robustness and competitive ranking performance across 11 tasks with only modest overhead over vanilla inference (Vazhentsev et al., 20 Feb 2025).
• Black-box entropy sampling with conformal prediction (TECP) yields reliable coverage and set-size tradeoffs without relying on logit access or auxiliary models (Xu, 30 Aug 2025).
• Conditional dependency correction methods (TAD) leveraging learned attention dependencies outperform baselines by 20–30 points in PRR for selective generation and hallucination rejection (Vazhentsev et al., 2024).

5. Applications Across Tasks and Modalities

Token-level uncertainty quantification underpins a wide spectrum of high-value tasks:

• Contextual Question Answering: Feature-gap UQ establishes state-of-the-art rejection and selection curves for both in-distribution and out-of-domain questions (Bakman et al., 3 Oct 2025).
• Fact-checking: Claim-Conditioned Probability (CCP) isolates semantic uncertainty in claim tokens, outperforming raw entropy, max-probability, and self-querying for fine-grained detection of unsupported statements (Fadeeva et al., 2024).
• Selective Generation and Cascading: Token-level uncertainty supports learned deferral in LM cascades, mitigating length bias and improving cost-quality tradeoffs by identifying hard instances requiring escalation (Gupta et al., 2024).
• Mathematical Reasoning: Epistemic uncertainty metrics directly correlate with correctness and guide the selection of high-quality solutions in multi-step compositions (Zhang et al., 16 May 2025).
• Dialogue and Embodied AI: Token-level p(action) or entropy scores provide conformal prediction-based coverage guarantees for safe action selection in interactive agents (Shorinwa et al., 2024).

6. Current Limitations and Open Research Problems

Despite technical advances, challenges remain:

• Semantic misalignment: Token entropy and related proxies do not consistently track factually correct outcomes, motivating continued research into semantic-decomposition methods and structured uncertainty (Shorinwa et al., 2024, Fadeeva et al., 2024).
• Prompt manipulation risk: Token-level uncertainty can be adversarially suppressed by prompt engineering or jailbreaks, leading to underreported uncertainty (Shorinwa et al., 2024).
• Scalability and interpretability: Methods relying on hidden-state geometric structure or batch-based centroids may need adaptation for very large models, multilingual settings, or multi-hop inference (Vazhentsev et al., 20 Feb 2025, Zur et al., 6 Nov 2025).
• Closed-source model opacity: White-box UQ is infeasible when logits/internal states are not exposed; black-box techniques (e.g., conformal prediction, output self-consistency) become necessary, often at a higher computational cost (Xu, 30 Aug 2025).
• Benchmarking and standardization: There is a lack of established per-token UQ benchmarks correlating uncertainty with downstream factual error rates beyond reading comprehension (Shorinwa et al., 2024).
• Conditional and interactive adaptation: Most methods focus on isolated generations, while in multi-turn or interactive settings, conditioning on uncertainty history and cross-episode calibration present unsolved challenges (Shorinwa et al., 2024).

7. Future Directions

Active lines of research include integrating token-level UQ with mechanistic interpretability (e.g., via probing of internal circuits or sparse autoencoders), leveraging latent uncertainty representations from hidden activations for global outcome forecasting (Zur et al., 6 Nov 2025), and extending density-based and causal feature models in multilingual or multimodal contexts. Conformal prediction, continuous semantic calibration, and context/history-aware UQ are prominent frontiers for both methodology and application development. Addressing these open problems is central to reliably quantifying epistemic uncertainty, mitigating hallucinations, and ensuring trustworthy deployment of LLMs across open-ended, high-stakes domains.