Entropy-Based Uncertainty Estimation

Updated 9 December 2025

Entropy-based uncertainty estimation is a method that uses various entropy measures (e.g., Shannon, Rényi, von Neumann) to quantify prediction uncertainty in probabilistic systems.
It extends classical approaches to capture nuanced uncertainties, including semantic, aleatoric, and epistemic components, enhancing risk assessment in complex models.
It underpins applications in LLMs, trajectory prediction, and quantum measurements, with advanced techniques like kernel language entropy and word-sequence entropy improving performance metrics.

Entropy-based uncertainty estimation refers to a family of methodologies that leverage formal entropy measures to quantify the degree of uncertainty associated with probabilistic or generative systems. Such measures have been deployed in diverse fields, from deep learning and LLMs to quantum physics, evidence theory, statistical signal processing, and dynamical systems. The foundational principle is that entropy, in its many guises (Shannon, Rényi, von Neumann, t-entropy, etc.), quantifies the unpredictability or spread in a system's state or predictions. Recent research extends classical entropy-based approaches to address nuanced forms of uncertainty that arise in high-dimensional, structured, or semantically-rich environments.

1. Mathematical Foundations of Entropy-Based Uncertainty

The canonical form of entropy-based uncertainty estimation is the computation of Shannon entropy for a discrete or continuous probability distribution. For a distribution $P = \{p_i\}$ , the Shannon entropy is given by

$H(P) = -\sum_{i} p_i \log p_i$

and quantifies the expected information content, or unpredictability, of random draws from $P$ . Extensions and generalizations include:

Rényi entropy for order $\alpha \neq 1$ :

$H_\alpha(P) = \frac{1}{1 - \alpha} \log \left( \sum_{i} p_i^\alpha \right)$

which emphasizes rare or frequent events depending on $\alpha$ (Lin et al., 6 Nov 2025).

von Neumann entropy for a density matrix $K$ (as used for positive semidefinite kernels):

$S(K) = -\mathrm{Tr}[K \log K]$

serving as a spectral generalization of Shannon entropy (Nikitin et al., 30 May 2024).

Approximate entropy (ApEn) uses methods from nonlinear time series analysis to quantify irregularity or incompleteness, particularly for network- or graph-based representations (Zhan et al., 2021).
t-Entropy, a bounded, concave measure based on the arctangent function, provides robust uncertainty estimates sensitive to rare events, with unique mathematical properties (Chakraborty et al., 2021).
Entropy power and its variants relate differential entropy of continuous distributions to equivalent Gaussian variances, yielding information-theoretic uncertainty relations in quantum settings (Jizba et al., 2016).
Mutual information (as a difference of entropies) naturally decomposes total uncertainty into aleatoric (inherent) and epistemic (model/reducible) components (Distelzweig et al., 2 Oct 2024).

2. Methodological Expansions: Beyond Predictive Entropy

Classical entropy of a predictive output distribution conflates uncertainty from all sources—syntactic, semantic, and statistical. However, high-stakes predictive systems require more precise quantification:

Semantic uncertainty quantification: Recent frameworks in LLMs (Kernel Language Entropy, KLE) recognize the need to measure uncertainty over meaning rather than surface forms. KLE constructs a positive semidefinite, unit-trace kernel $K$ encoding pairwise semantic similarity between sampled outputs, then computes the von Neumann entropy on the spectrum of $K$ , thereby capturing graded and cluster-relations among outputs:

$\mathrm{KLE}(x) = -\mathrm{Tr}[K \log K],\;\; K_{ij} = k(S_i, S_j),\; \mathrm{Tr} K = 1$

This strictly generalizes hard-cluster-based “semantic entropy” and allows for continuous distinctions that discrete clustering erases (Nikitin et al., 30 May 2024).

Sample-based and label-confidence-aware uncertainty: Bayesian and frequentist generative models often estimate uncertainty from samples. Advanced approaches address the bias introduced by using greedy (argmax) outputs as reference labels, and correct it by augmenting entropy-based uncertainty with Kullback-Leibler divergence between the sample distribution and the label-source distribution (label-confidence-aware uncertainty) (Lin et al., 10 Dec 2024).
Aleatoric and epistemic uncertainty decomposition: In Bayesian inference and deep ensemble methods for tasks such as trajectory prediction, entropy-based uncertainty is decomposed as

$\mathrm{Total} = H[y|x,\mathcal{D}] = \mathbb{E}_{p(w|\mathcal{D})}[H[y|x,w]] + I[y,w|x]$

where the first term is the (data) expected entropy (aleatoric) and the second is mutual information (epistemic) (Distelzweig et al., 2 Oct 2024).

Graph- and kernel-based uncertainty: For outputs with relational or latent structure, entropy over kernels (formed via graph Laplacians, heat or Matérn kernels on similarity graphs) integrates semantic distance and uncertainty in a principled spectral fashion (Nikitin et al., 30 May 2024).
Relevance-weighted entropy: Entropy scores can be modulated by semantic relevance at word and sequence levels, as in Word-Sequence Entropy (WSE), which weights token-level entropy by learned or model-inferred importance, mitigating the effects of irrelevant “generative inequality” in open-ended response spaces (Wang et al., 22 Feb 2024).

3. Engineering and Application Domains

Entropy-based uncertainty estimation underpins workflows in a wide range of domains:

Domain	Entropy/Uncertainty Role	Representative Formulation
LLMs	Semantic hallucination/error detection, answer trust calibration	Kernel Language Entropy, Semantic Entropy (Nikitin et al., 30 May 2024, Sharma et al., 17 Feb 2025)
Code Generation	Abstention policies, correctness proxies	Semantic entropy over equivalence clusters (Sharma et al., 17 Feb 2025)
Trajectory Prediction	Safety in autonomous systems, OOD detection	Predictive/aleatoric/epistemic decomposition (Distelzweig et al., 2 Oct 2024)
Speech and Signal Processing	SNR estimation via DNN output entropy and Bayesian dropout variance	Framewise Shannon entropy, MC-dropout variance (Aralikatti et al., 2018)
Medical Image Segmentation	Error flagging under domain shift, calibration	Per-voxel Shannon entropy, entropy regularization (Matzkin et al., 17 Jun 2025)
Evidence Theory	Completeness of basic probability assignments	Approximate entropy on degree sequence (Zhan et al., 2021)
Quantum Measurement	Entropic uncertainty relations for POVMs, tomography	Shannon entropy bounds via Taylor/Chebyshev expansions (Rastegin, 2020, 1105.4865)
Statistical Inference	Plug-in and bootstrap uncertainty/c.i. for nonparametric entropy	Entropy via mixture models with WLB intervals (Robin et al., 2020, Scrucca, 27 May 2024)

In these contexts, entropy-based uncertainty estimation is valued for both its theoretical interpretability (information-theoretic, Bayesian, or analytic underpinnings) and its practical effect on system reliability.

4. Theoretical Properties, Estimation, and Error Bounds

Several key theoretical properties and estimation frameworks have emerged:

Spectral/universal form: Entropy of kernel spectra reduces to Shannon entropy for eigenvalue pmf; in the limit, KLE generalizes to standard hard clustering-based methods (Nikitin et al., 30 May 2024).
Statistical efficiency: Harmonic number-based and mixture-based plug-in entropy estimators attain asymptotically efficient variance under sample size and distributional tail constraints, enabling confidence intervals and robust large-alphabet inference (Mesner, 26 May 2025, Scrucca, 27 May 2024).
Bootstrap and credibility intervals: The uncertainty of entropy estimates can be itself quantified, e.g., via weighted likelihood bootstrap (with Dirichlet weights tuned for frequentist coverage) on mixture models, which outperforms classical nonparametric bootstrap and percentile intervals for entropy uncertainty (Scrucca, 27 May 2024).
High-order bounds on uncertainty: In quantum, stochastic, and active matter systems, higher-order uncertainty relations (thermodynamics, quantum), built on entropy or cumulant generating functions, offer tight bounds where traditional variance-based measures fail, particularly in far-from-equilibrium or heavy-tailed regimes (Bao et al., 2022, Jizba et al., 2016).
Concavity, invariance, robustness: Novel entropy measures such as t-entropy are concave, bounded, and continuous, and preserve majorization orderings and robustness to rare events, which is not guaranteed for Shannon or Rényi entropy in all settings (Chakraborty et al., 2021).

5. Empirical Performance and Benchmarking

Comprehensive empirical assessments demonstrate the superiority or complementary nature of advanced entropy-based uncertainty estimators:

LLMs & NLG: KLE with graph-heat kernels outperforms pure semantic entropy in AUROC metrics for distinguishing correct from hallucinated answers across ~75% of model-dataset scenarios, is robust to black-box evaluation settings, and retains performance under default kernel parameter choices (Nikitin et al., 30 May 2024). WSE-based posterior filtering enables 2–40 percentage point improvements in accuracy of selected low-uncertainty answers across diverse LLMs and datasets (Wang et al., 22 Feb 2024). Label-confidence-aware entropy yields >10 point average AUROC gains compared to naive entropy across free-form QA tasks and models (Lin et al., 10 Dec 2024).
Code generation: Abstention thresholds on uniform or length-normalized functional semantic entropy reduce unsafe code emission rates to near zero without significantly sacrificing correct outputs (Sharma et al., 17 Feb 2025).
Trajectory and motion prediction: Entropy/mutual information decomposition yields high correlation with realized prediction error (minADE), spikes in epistemic uncertainty robustly indicate OOD settings, and deep ensembles consistently yield strongest uncertainty quality (Distelzweig et al., 2 Oct 2024).
Statistical estimation: Mixture-model plug-in entropy estimates, when coupled with Dirichlet WLB quantification, achieve nominal or better 95% empirical coverage for uncertainty, and harmonic number estimators yield asymptotic normality with minimax mean squared error on heavy-tail discrete distributions (Mesner, 26 May 2025, Scrucca, 27 May 2024).

6. Limitations, Open Problems, and Future Directions

While entropy-based uncertainty estimation provides theoretically sound and practically effective mechanisms, several limitations and ongoing research directions are evident:

Computational overhead: Construction of kernel matrices (e.g., KLE) and their eigendecomposition is cubic in sample count but manageable for pragmatic output sizes ( $n=10$ in LLM settings) (Nikitin et al., 30 May 2024). Semantic similarity assessments (WSE, KLE) involving NLI or cross-encoders induce non-trivial latency and potential biases due to external model imperfections (Wang et al., 22 Feb 2024).
Transfer to real-time and multimodal tasks: Sampling costs may challenge deployment in real-time or resource-constrained applications; future work aims at sample reduction (e.g., via projective clustering), adaptive kernel learning per input, or direct multimodal extensions (vision, code, etc.) (Nikitin et al., 30 May 2024, Sharma et al., 17 Feb 2025).
Label-source mismatch: Biases introduced by greedy or outlier-prone decodings motivate continued integration of confidence-aware and sample-distribution-calibrated uncertainty mechanisms (Lin et al., 10 Dec 2024).
Completeness and epistemic uncertainty: Evidence theory measures such as ApEn-based “integrity” are sensitive to lack of completeness but not semantic “incorrectness”; complementary approaches (e.g., fusion with epistemic mutual information) may be needed for robust decision-making (Zhan et al., 2021).
Design of bounded or robust measures: Measures like t-entropy and bounded divergences offer increased noise resistance and breakdown robustness; extending these to structured or streaming settings is an ongoing topic (Chakraborty et al., 2021).

7. Synthesis and Outlook

Entropy-based uncertainty estimation offers a mathematically mature and empirically validated framework for quantifying uncertainty in complex prediction, inference, and decision systems. Progress over the last decade extends beyond naïve predictive entropy to address semantic uncertainty, sample-level calibration, robustness, and application to non-classical probability models. Advances such as Kernel Language Entropy (Nikitin et al., 30 May 2024), Word-Sequence Entropy (Wang et al., 22 Feb 2024), and entropy-regularized inference (Kaur, 15 Mar 2025) exemplify the direction toward invertible, task-sensitive, and theoretically sound uncertainty measures, sensitive both to distributional richness and operational semantics. These developments play a critical role in elevating the trustworthiness, interpretability, and adaptability of modern AI and statistical systems.