Rethinking Uncertainty Estimation in LLMs: A Principled Single-Sequence Measure¶

Conference: ICLR 2026 arXiv: 2412.15176 Code: None Area: Text Generation / Uncertainty Estimation Keywords: uncertainty estimation, greedy decoding, negative log-likelihood, proper scoring rules, LLM

TL;DR¶

Starting from the proper scoring rules framework, this paper proves that the negative log-likelihood of the highest-probability output sequence (MSP) is a theoretically grounded uncertainty measure, and proposes G-NLL — a method that approximates this measure with a single greedy decoding pass, matching or surpassing SOTA methods that require multiple samples across several benchmarks.

Background & Motivation¶

Background: LLM uncertainty estimation primarily relies on the logarithmic scoring rule, yielding measures such as predictive entropy (PE) and semantic entropy (SE) that require sampling multiple output sequences for approximation, incurring substantial computational cost.

Limitations of Prior Work: Multi-sequence sampling methods are impractical for real-world deployment — sampling 10 sequences implies a 10× inference overhead. Furthermore, differences among sampled sequences may reflect mere lexical variation rather than genuine uncertainty, necessitating additional natural language inference models for semantic clustering, further increasing complexity.

Key Challenge: The logarithmic scoring rule inherently requires computing an expectation over the entire output distribution (Shannon entropy), which grows exponentially with sequence length and is fundamentally intractable to compute exactly. The question arises: is there a proper scoring rule that does not require exhaustive enumeration of the distribution?

Goal: (a) Provide a theoretical foundation for single-sequence uncertainty measures; (b) analyze the sampling complexity advantages of the proposed approximation; (c) deliver the most efficient practical implementation.

Key Insight: Explore the zero-one score as an alternative proper scoring rule. Under this rule, aleatoric uncertainty depends solely on the likelihood of the highest-probability sequence, eliminating the need for full-distribution sampling.

Core Idea: Replace the logarithmic scoring rule with the zero-one scoring rule to derive an uncertainty measure, revealing that only the negative log-likelihood of the greedily decoded sequence is required.

Method¶

Overall Architecture¶

Given an LLM and a prompt \(\bm{x}\), the method outputs an uncertainty estimate for that prompt. Unlike existing approaches that sample \(N\) sequences and compute aggregate statistics, G-NLL requires only a single greedy decoding pass: taking the argmax at each token position and accumulating the negative log-probabilities.

Key Designs¶

Zero-One Scoring Rule Derivation of MSP:
Function: Derive the maximum sequence probability (MSP) as an aleatoric uncertainty measure from the proper scoring rules framework.
Mechanism: \(\mathbf{S}_{0\text{-}1}(p, \bm{y}') = (1 - p(\bm{y}=\bm{y}'|\bm{x})) \cdot \mathbb{1}\{\bm{y}'=\arg\max p(\bm{y}|\bm{x})\}\). Substituting into the decomposition formula, the aleatoric uncertainty reduces to \(1 - p(\bm{y}=\bm{y}^*|\bm{x},\bm{w})\), i.e., MSP, which depends only on the highest-probability sequence.
Design Motivation: The logarithmic scoring rule requires an expectation over all possible outputs (on the order of \(|\mathcal{V}|^T\)), whereas the zero-one scoring rule only requires identifying the highest-probability sequence, fundamentally reducing computational demand.
G-NLL Approximation:
Function: Approximate the highest-probability sequence via greedy decoding.
Mechanism: \(\text{G-NLL} = -\sum_{t=1}^T \log(\max_{y_t} p(y_t|\bm{x}, \bm{y}_{<t}, \bm{w}))\), which decomposes the sequence-level max into a token-level greedy max. Although the greedy sequence is not guaranteed to be the globally optimal sequence, both experiments and theoretical simulations demonstrate that the approximation quality is sufficient.
Design Motivation: Exactly finding the highest-probability sequence still requires searching an exponential space, whereas greedy decoding is part of the standard inference pipeline and introduces zero additional overhead.
Sampling Complexity Theoretical Analysis (Theorem 1):
Function: Prove that the sampling complexity for approximating \(M(p(\bm{y}))\) is substantially lower than for approximating \(H(p(\bm{y}))\).
Mechanism: The sampling complexity for approximating \(M\) is \(O(\frac{C_\epsilon}{P_\epsilon}\log\frac{1}{\delta})\), depending on the probability mass concentration in the \(\epsilon\)-neighborhood. The complexity for approximating \(H\) is \(O(\frac{(b-a)^2 C^2}{2\epsilon^2}\log\frac{2}{\delta})\), depending on the likelihood range and worst-case importance weights.
Design Motivation: The concentrated nature of LLM output distributions makes \(M\) easy to approximate, while approximating \(H\) is considerably more difficult.

Loss & Training¶

G-NLL requires no training and is a purely inference-time method. A key finding is that length normalization should not be applied to G-NLL — normalization breaks the theoretical correspondence with MSP.

Key Experimental Results¶

Main Results (6 models × 6 tasks, AUROC for distinguishing correct/incorrect answers)¶

Six language models spanning different architectures (transformer, state-space), sizes (7B, 8B, 70B), and training stages (PT, IT):

Method	# Sampled Sequences	Avg. AUROC	Notes
PE	10	Baseline	Predictive entropy
LN-PE	10	Slightly higher	Length-normalized PE
SE	10	Moderate	Semantic entropy
D-SE	10	Moderate	Improved semantic entropy
G-NLL	1	SOTA	10× efficiency gain

Ablation Study¶

Configuration	Performance	Notes
G-NLL (no normalization)	Best	Theoretically correct form
G-NLL + length normalization	Degraded	Breaks correspondence with MSP
Sampled sequence NLL (non-greedy)	Degraded	Theoretical guarantee holds only for the highest-probability sequence
PE (\(N=5\))	Degraded	Too few samples, high variance

Key Findings¶

G-NLL with 1 decoding pass matches or exceeds PE/SE with 10 samples — 10× computational efficiency gain.
Length normalization should not be applied to G-NLL; it lacks theoretical justification and is empirically harmful.
Greedy decoding (highest-probability sequence) is essential; NLL of sampled sequences performs worse.
Simulation experiments show that the approximation error of greedy decoding to MSP is far smaller than that of multi-sequence sampling to PE.
G-NLL performs consistently across different model architectures and scales.

Highlights & Insights¶

The theoretical contribution is the central highlight: This work provides the first proper scoring rule foundation for single-sequence uncertainty measures (MSP), elevating a previously ad hoc baseline into a theoretically principled method. This challenges the prevailing assumption that multi-sequence sampling is necessary for reliable uncertainty estimation.
High practical value: G-NLL is simply the negative log-likelihood of the greedy decoding output, incurring zero additional computational cost and serving directly as an uncertainty signal in LLM deployment.
The sampling complexity analysis provides a theoretical benchmark for the computational feasibility of different uncertainty measures.

Limitations & Future Work¶

Greedy decoding does not necessarily find the true highest-probability sequence (which is NP-hard), and serves only as an upper-bound approximation.
The paper focuses solely on aleatoric uncertainty and does not address epistemic uncertainty.
Experiments are limited to question-answering tasks; the method has not been validated in long-form text generation settings.
The semantic-level counterpart of the zero-one score (MCP) remains insufficiently explored.

vs. PE (Malinin & Gales): PE is based on the Shannon entropy of the logarithmic scoring rule, requiring multiple samples with high variance. G-NLL is based on the zero-one scoring rule and requires only a single decoding pass.
vs. SE (Kuhn et al.): SE further introduces semantic clustering to reduce spurious uncertainty but requires an additional NLI model. G-NLL requires no auxiliary models.
vs. Fadeeva et al.: They proposed MSP as a baseline without theoretical justification; this paper supplies the missing theoretical foundation.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ Challenges the dominant paradigm from a theoretical standpoint; elegant and practical.
Experimental Thoroughness: ⭐⭐⭐⭐ 6 models × 6 tasks, with simulation analyses and theoretical proofs.
Writing Quality: ⭐⭐⭐⭐⭐ Rigorous and clear theoretical derivations; compelling research motivation.
Value: ⭐⭐⭐⭐⭐ Has paradigm-shifting implications for LLM uncertainty estimation.