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

Large Language Models (LLMs) are increasingly employed in real-world applications, driving the need to evaluate the trustworthiness of their generated text. To this end, reliable uncertainty estimation is essential. Leading uncertainty estimation methods generate and analyze multiple output sequences, which is computationally expensive and impractical at scale. In this work, we inspect the theoretical foundations of these methods and explore new directions to enhance computational efficiency. Building on the framework of proper scoring rules, we find that the negative log-likelihood of the most likely output sequence constitutes a theoretically principled uncertainty measure. To approximate this alternative measure, we propose G-NLL, obtained using a single output sequence from greedy decoding. This approach streamlines uncertainty estimation while preserving theoretical rigor. Empirical results demonstrate that G-NLL achieves state-of-the-art performance across various scenarios. Our work lays the theoretical foundation for efficient and reliable uncertainty estimation in natural language generation, challenging the necessity of the prevalent methods that are more complex and resource-intensive.

💡 Research Summary

The paper addresses the growing need for reliable uncertainty estimation in large language models (LLMs) deployed in real‑world applications. Existing approaches typically rely on sampling multiple output sequences and computing measures such as predictive entropy or semantic entropy, which require expectations over the full output distribution. Because the space of possible sequences grows exponentially with vocabulary size and sequence length, these methods are computationally prohibitive and often provide noisy signals—different token‑level outputs can be semantically identical, inflating entropy‑based uncertainty.

To overcome these limitations, the authors revisit the theory of proper scoring rules, a class of loss functions that are minimized when the predicted distribution matches the true distribution. The standard logarithmic score leads to the negative log‑likelihood (NLL) of a generated sequence and, when applied to NLG, yields uncertainty expressed as the entropy of the full sequence distribution plus a KL‑divergence term. Both terms require summing over all possible sequences, which in practice is approximated by Monte‑Carlo sampling, preserving the need for many generations.

The key insight of the paper is to replace the logarithmic score with the zero‑one score. The zero‑one score only cares about the probability assigned to the most likely sequence (the MAP sequence). Substituting this score into the proper‑scoring‑rule framework collapses the uncertainty measure to a simple function of the probability of the MAP sequence:

 Uncertainty = 1 − p(y* | x)

where y* = arg max_y p(y | x). This quantity is exactly the “Maximum Sequence Probability” (MSP) that has appeared in recent empirical work but lacked a solid theoretical justification. The authors formally derive MSP as a strictly proper scoring‑rule‑based uncertainty measure, establishing its statistical soundness.

Finding the exact MAP sequence is NP‑hard for modern LLMs, so the paper proposes a practical approximation: use greedy decoding (also called greedy or argmax decoding) to obtain a single output sequence and its log‑likelihood. The resulting metric, called G‑NLL (Greedy Negative Log‑Likelihood), is computationally cheap—it requires only one forward pass and no sampling. Despite this simplicity, G‑NLL retains the theoretical guarantees of the underlying zero‑one‑score‑based uncertainty.

Extensive experiments evaluate G‑NLL across a wide spectrum of models (7 B to 70 B parameters), training stages (pre‑training vs. fine‑tuning), tasks (question answering, summarization, translation, code generation), and datasets (TruthfulQA, SQuAD, XSum, WMT). The authors compare against strong baselines: Predictive Entropy (PE), Semantic Entropy (SE), Monte‑Carlo dropout, deep ensembles, and other recent single‑sequence baselines. Evaluation metrics include ROC‑AUC for error detection, Expected Calibration Error (ECE), Brier score, and direct correlation with factual errors.

Results show that G‑NLL matches or exceeds the performance of all baselines on uncertainty detection while being orders of magnitude faster. In particular, G‑NLL avoids the need for length normalization, a common source of instability for likelihood‑based measures, and it scales linearly with decoding time. Beam‑search approximations of MSP improve accuracy only marginally but increase computational cost dramatically, confirming the efficiency‑effectiveness trade‑off.

The authors also discuss limitations. Greedy decoding may miss the true MAP sequence, leading to under‑estimation of uncertainty when the model’s top‑probability path is not greedy. However, empirical analysis indicates that the gap between greedy and beam‑search probabilities is small for the models studied. Moreover, the zero‑one‑score formulation assumes that the most probable sequence is the correct answer; systematic model biases could therefore cause over‑confidence. The paper suggests future work combining G‑NLL with calibration techniques or hybrid schemes that evaluate a small set of high‑probability candidates.

In summary, the paper makes three primary contributions: (1) a rigorous derivation of the maximum sequence probability as a proper‑scoring‑rule‑based uncertainty measure for NLG; (2) a theoretical and empirical analysis showing that single‑sequence measures possess desirable properties compared to multi‑sample methods; (3) the introduction of G‑NLL, a greedy‑decoding‑based approximation that delivers state‑of‑the‑art uncertainty estimation with negligible computational overhead. This work challenges the prevailing belief that multiple sampled outputs are necessary for reliable uncertainty quantification in LLMs and provides a practical tool for deploying trustworthy language models at scale.