Token-Entropy Conformal Prediction (TECP)

Updated 6 September 2025

TECP is a statistical framework that uses token-level entropy to quantify uncertainty in open-ended language generation by black-box LLMs.
It employs split conformal prediction to calibrate token-level nonconformity scores, ensuring that prediction sets meet formal coverage guarantees.
Empirical results show that TECP achieves robust error control and compact prediction sets across various LLMs under different evaluation metrics.

Token-Entropy Conformal Prediction (TECP) is a statistical framework for uncertainty quantification in open-ended natural language generation tasks, particularly LLMs operating under black-box constraints. TECP leverages token-level entropy—computed solely from generated text—to produce prediction sets with formal coverage guarantees. This approach provides a principled and model-agnostic alternative to methods reliant on semantic consistency heuristics or internal model features, enabling robust error control and efficient prediction set construction for trustworthy language generation.

1. Conceptual Foundations

Token-Entropy Conformal Prediction (TECP) is motivated by the challenge of quantifying epistemic uncertainty in open-ended LLM outputs, especially in scenarios where internal scores (such as logits or attention weights) are inaccessible. TECP employs token-level entropy as a direct, logit-free, reference-free measure of uncertainty; the cumulative entropy of a generated sequence serves as the nonconformity score in a split conformal prediction pipeline. Given a set of candidate outputs from an LLM (typically generated via stochastic decoding such as temperature sampling or beam search), TECP computes the total entropy over all tokens in each sequence, assessing the degree of model uncertainty inherent in each generated answer. The integration of this nonconformity score into a formal conformal prediction pipeline ensures that constructed prediction sets—subsets of candidate answers for each input—achieve rigorous coverage guarantees.

2. Mathematical Formulation

Let $x$ denote the input (e.g., a natural language question), and $\mathcal{Y}(x) = \{\hat{y}_1, \ldots, \hat{y}_M\}$ the set of $M$ candidate outputs generated by an LLM for $x$ . For a candidate sequence $\hat{y}_m$ of length $L_m$ , the token-entropy nonconformity score is defined as:

$U(\hat{y}_m) = \sum_{t=1}^{L_m} H_t$

$H_t = -\sum_{v \in \mathcal{V}} p_t(v) \log p_t(v)$

where $p_t(v)$ is the sampled probability of token $v$ at position $t$ , and $\mathcal{V}$ is the model’s vocabulary.

To calibrate uncertainty thresholds, TECP uses split conformal prediction. The available dataset is filtered to ensure at least one semantically correct candidate per input. Then, the filtered samples are randomly divided into a calibration set and a test set. For the calibration set, nonconformity scores $U(\hat{y}_m)$ are collected for all semantically correct candidates (as determined by reference matching). The conformal quantile $\hat{q}_{\alpha}$ is computed as:

$\hat{q}_{\alpha} = \text{Quantile}(\mathcal{R}, q\_level)$

$q\_level = \frac{\lceil (1 - \alpha) (n + 1) \rceil}{n}$

where $\mathcal{R}$ is the multiset of calibration scores and $n$ its cardinality. For any input $x$ in the test set, the TECP prediction set is

$\Gamma(x) = \{\hat{y}_m \in \mathcal{Y}(x) : U(\hat{y}_m) \leq \hat{q}_{\alpha}\}$

This set construction guarantees (under exchangeability assumptions) that the prediction set $\Gamma(x)$ includes at least one correct answer with probability at least $1 - \alpha$ .

3. Algorithmic Workflow

The TECP process consists of:

Candidate Generation: For each input $x$ (e.g., a question), $M$ candidate sequences are generated using an LLM with randomized decoding (e.g., via temperature sampling).
Entropy Computation: For each candidate $\hat{y}_m$ , sum token entropies across the sequence to yield $U(\hat{y}_m)$ .
Dataset Filtering: Retain only those examples where at least one candidate satisfies semantic equivalence with the ground truth.
Data Splitting: Partition the filtered dataset into calibration and test sets (random split).
Calibration: Aggregate $U(\hat{y}_m)$ for matching candidates from the calibration set; compute the conformal quantile $\hat{q}_{\alpha}$ based on the desired coverage.
Prediction Set Construction: For each test input, select candidates with $U(\hat{y}_m) \leq \hat{q}_{\alpha}$ to form $\Gamma(x)$ .
Coverage Guarantee: By properties of split conformal prediction, $\Pr\{\text{correct answer} \in \Gamma(x)\} \geq 1 - \alpha$ .

This workflow is entirely compatible with black-box models, requiring only access to output tokens and their associated probabilities at generation.

4. Empirical Performance and Evaluation

TECP is benchmarked on six modern LLMs—LLaMA-3.2-1B, Qwen2.5-3B-Instruct, Vicuna-7B-v1.5, Qwen2.5-7B-Instruct, LLaMA-3.1-8B-Instruct, and Vicuna-13B-v1.5—across TriviaQA and CoQA QA datasets. Two principal metrics are reported:

Metric	Definition	Interpretation
Expected Metric Recall (EMR)	Fraction of test inputs whose prediction set contains at least one sufficient candidate	Coverage quality
Average Prediction Set Size (APSS)	Average number of candidates per prediction set	Efficiency of uncertainty quantification

Under risk level $\alpha < 0.2$ , EMR scores for TECP commonly fall near 0.1 or better, indicating tight control of incorrect candidate inclusion. Larger models (Vicuna-13B) demonstrate higher selectivity and concentration, yielding smaller prediction sets. Compared to baseline methods relying on semantic self-consistency (ConU), TECP provides more stable risk calibration, lower prediction set size variance across random splits, and consistent adherence to theoretical risk bounds.

5. Black-Box Applicability and Model-Agnosticism

A distinguishing feature of TECP is its capacity for black-box uncertainty quantification—requiring only input-output access rather than internal model signals. In many production or API-based LLM deployments (e.g., ChatGPT), direct extraction of logits, attention, or other white-box features is unavailable. TECP’s reliance solely on token-level entropy calculated from sampled generations makes it a universally applicable solution. The entropy measure directly captures epistemic uncertainty inherent in the generation process rather than external semantic heuristics, resulting in robust quantification not easily susceptible to idiosyncratic biases.

6. Relation to Conformal Prediction Theory and Notions of Efficiency

TECP builds on the split conformal prediction paradigm, where a nonconformity score is transformed via empirical calibration to yield prediction sets with finite-sample guarantees. Traditional conformal prediction often utilizes error residuals, semantic consistency scores, or internal features; TECP’s use of entropy is distinctive, producing an uncertainty measure that reflects the probabilistic dispersion over token choices. The underlying statistical guarantee—coverage rate at least $1 - \alpha$ —is sustained by the quantile-based cutoff. Unlike calibration methods that may inflate prediction set size (cf. temperature scaling (Xi et al., 6 Feb 2024)), TECP provides coverage-conservative yet efficient prediction sets (lower APSS), thus supporting improved efficiency for open-ended language generation.

7. Advantages, Limitations, and Open Directions

Advantages:

Logit-free, reference-free UQ for black-box LLMs.
Formal distribution-free coverage guarantees via conformal calibration.
Empirically validated for reliability and compactness across diverse models and datasets.

Limitations:

TECP assumes reliable token entropy extraction, which may be affected by model-specific artifacts (instruction tuning, RLHF, etc.).
Coverage-control depends on robust semantic matching in filtering; ambiguous references may complicate set size calibration.
The method may yield conservative prediction sets (larger APSS) in low-confidence or highly uncertain settings; tuning $\alpha$ and candidate sampling may be required for optimal efficiency.

A plausible implication is that further research may refine TECP by introducing contextual or local weighting in the entropy aggregation (cf. localized conformal prediction (Guan, 2019)), adaptive efficiency constraints (cf. constrained ERM (Bai et al., 2022)), or by integrating auxiliary information for enhanced calibration (cf. information-theoretic bounds (Correia et al., 3 May 2024)).

8. Connections to the Broader Literature

TECP inherits statistical rigor from the conformal prediction literature, while extending its practical applicability to black-box settings and sequential prediction. Previous works demonstrate that calibration-attempts via post-hoc temperature scaling may inflate prediction sets (Xi et al., 6 Feb 2024), and entropy-based reweighting imbues efficiency improvements in classification (Luo et al., 24 Jul 2024). The information-theoretic perspective on prediction set size and entropy (Correia et al., 3 May 2024) supports TECP’s focus on token-level uncertainty as an explicit measure of epistemic unpredictability; efficient uncertainty quantification leads to smaller prediction sets and tighter error control.

In summary, TECP constitutes a formal, robust, and efficient approach to uncertainty quantification for open-ended language generation, unifying information-theoretic uncertainty with nonparametric calibration via split conformal prediction, and is empirically validated for contemporary LLMs under black-box constraints (Xu, 30 Aug 2025).