Entropy–Capacity Logprob-Native Inference

• Entropy–Capacity Logprob-Native Inference is a unified framework that combines maximum entropy principles, channel capacity, and thermodynamic analogies to enhance model inference.
• It leverages native log-probability signals to dynamically allocate model capacity and detect unreliable outputs, improving both accuracy and computational efficiency.
• The framework establishes fundamental capacity bounds and entropy inequalities that integrate ideas from statistical mechanics, Bayesian inference, and modern machine learning.

Entropy–Capacity Logprob-Native Inference describes a spectrum of information-theoretic frameworks and mechanisms that jointly quantify uncertainty (entropy) and the ability of modeling resources or evidence (capacity) to constrain inference, with native dependence on explicit log-probabilities. These methods provide unified formulations for probabilistic updating, detection of unreliable outputs, dynamic allocation of model capacity, and principled channel capacity bounds. Foundational results connect concepts from statistical mechanics, Bayesian inference, information theory, and modern machine learning.

1. Theoretical Foundations: Entropy, Capacity, and Inference

At its core, entropy–capacity logprob-native inference unifies the logic of rational inference (the ME or Maximum Entropy method), formal channel capacity, and thermodynamic analogies:

• Maximum Entropy (ME) Method: The ME framework uniquely identifies the logarithmic relative entropy as the tool for updating from prior $q(x)$ to posterior $p(x)$, selecting $p$ to maximize $S[p,q] = -D[p\|q]$, subject to new constraints. The uniqueness of KL divergence follows from the requirements of subdomain locality and subsystem independence (Caticha, 2021).
• Capacity: In Shannon theory, capacity quantifies the maximum reliable rate at which information can be transmitted given noise and input constraints. Within the ME paradigm, independence—a privileged assumption—directly supports local, modular inference and is mirrored by the capacity metric in communication.
• Thermodynamic/Information Theory Analogy: Statistical physics provides an isomorphism: free energy maps to log-evidence, heat capacity to learning capacity, and Gibbs entropy to the effective number of distinguishable models. These identifications form the statistical mechanics of inference (LaMont et al., 2017).

2. Entropy–Capacity Principles in Logprob-Native LLM Inference

Modern instantiations leverage token-level log-probabilities as primary information sources, enabling mechanisms such as ECLIPSE and Entropy Adaptive Decoding (EAD) to operationalize entropy–capacity objectives:

• ECLIPSE Framework (Singha, 2 Dec 2025): Hallucination is treated as a misalignment where a LLM's semantic entropy $H_\text{sem}$ overshoots the “evidence capacity” $C_\text{evd}$ provided by retrieved input. $H_\text{sem}$ is estimated by clustering outputs from sampled generations, while $C_\text{evd}$ is the difference in log-probabilities for the top answer, with and without evidence:

$$C_\text{evd} = w_\text{cons} \big[ \log p_\theta(A^*|Q,E) - \log p_\theta(A^*|Q) \big]$$

This approach exploits logprob-native signals such as $L_{QE}$ (log-likelihood given evidence), $L_Q$ (log-likelihood without evidence), $\Delta L = L_{QE} - L_Q$, and auxiliary metrics based on the entire token sequence.

• Entropy Adaptive Decoding (EAD) (Simonds, 5 Feb 2025): Dynamic inference efficiency is achieved by monitoring rolling entropy in token-level model predictions, switching computation between small and large models as local uncertainty (entropy) crosses a threshold $\tau$. The construction is inherently logprob-native: no confidence proxies or external calibration mechanisms are required, and the actual softmax probabilities over tokens drive decision making.

3. Methodological Implementation and Logprob-Decomposition Features

Both ECLIPSE and EAD rely on direct access to model log-probabilities, enabling precise entropy and capacity computation:

• Semantic Entropy Calculation: Multi-sample clustering of generated sequences is performed; clusters represent meaning-equivalent answers, and entropy is computed over empirical cluster probabilities.
• Perplexity Decomposition: Explicit log-probabilities for answer strings are computed with and without evidence (retrieval augmentation), yielding feature vectors $$(\hat H_\text{sem}, C_\text{evd}, L_Q, L_{QE}, \Delta L, \text{ratio}, p_\text{max})$$ which provide the basis for robust hallucination detection (Singha, 2 Dec 2025).
• Dynamic Model Switching: EAD tracks Shannon entropy over a fixed window of tokens during generation; once average entropy exceeds a preset threshold, the system seamlessly transitions to a higher-capacity model, with computational savings tunable via $\tau$.

Feature	Role	ECLIPSE/EAD Use
$H_\text{sem}$	Output entropy	Hallucination detection
$C_\text{evd}$	Evidence capacity	Hallucination detection
$L_{QE}, L_Q, \Delta L$	Capacity lift	Feature importances
Rolling entropy	Uncertainty	Model switching

Empirical analyses demonstrate that the largest and most-significant coefficients in trained detectors arise not from entropy alone, but from the interplay with evidence capacity and logprob-native difference features, especially $\Delta L$ and $L_{QE}/L_Q$.

4. Channel Capacity, Entropy Inequalities, and Log-Concave Distributions

Capacity results for stochastic channels with log-concave noise directly invoke entropy–capacity logic in settings constrained by hard physical or statistical rules:

• Sharp Capacity Bounds: For additive noise channels with symmetric log-concave noise $N$ (variance $\sigma^2_N$), the achievable capacity $C(P)$ under input power constraint $P$ is at most $0.254$ bits above that of the Gaussian noise channel with identical noise power:

$C(P) \leq C_\mathcal{G}(P) + 0.254~\text{bits}$

with $$C_\mathcal{G}(P) = \frac{1}{2} \log_2 (1 + P/\sigma_N^2)$$ (Madiman et al., 2018).

• Moment-Entropy Inequalities: The minimum differential entropy for symmetric log-concave distributions at fixed variance is uniquely achieved by the uniform distribution, and the maximizer is Gaussian—mirroring the classic entropy–capacity correspondence.

These results illustrate the fundamental role of entropy–capacity matching in bounding achievable rates and in quantifying the “efficiency gap” induced by departure from optimal (e.g., non-Gaussian) channel noise.

5. Thermodynamic Analogies, Learning Capacity, and Inference Geometry

The analogy with statistical physics formalizes the relation between entropy, capacity, and "degrees of freedom" in both learning and inference:

• Learning Capacity: The analog of heat capacity, defined as

$$C(N) = N^2 \, \mathrm{Var}_{\pi(\theta|x^N)}[\hat H(\theta)]$$

captures the rate at which expected predictive loss decreases with increasing sample size. Regular models reach $C \rightarrow K/2$ asymptotically, where $K$ is the parameter dimension (LaMont et al., 2017).

• Gibbs Entropy in Bayesian Inference: The effective count of distinguishable parameter values, $e^{S_G}$, contracts as data accumulate; $S_G$ quantifies the “active capacity” remaining for further learning.
• Objective Priors and Indifference: The generalized principle of indifference (GPI) selects priors to assign equal probability to each distinguishable distribution, yielding priors that adapt to sample size and model structure.

This framework clarifies the penalization mechanism for model complexity and links the overfitting tendencies to learning capacity rather than to ad hoc complexity metrics.

6. Applications and Empirical Evidence

• Detection of Model Hallucinations: ECLIPSE achieves an ROC AUC of 0.89 in financial QA hallucination detection, versus 0.50 for an entropy-only baseline; performance collapses when logprob-native features are ablated or replaced with proxies (Singha, 2 Dec 2025).
• Accelerated Inference in LLMs: EAD maintains over 92% of a high-capacity model’s performance while reducing compute by over 60%, as measured by the proportion of tokens generated by large versus small models, with a controlled tradeoff between oracle fidelity and resource expenditure (Simonds, 5 Feb 2025).
• Channel Coding Theory: Capacity bounds connect with real-world limits for noise-tolerant transmission and timing channels in the presence of log-concave and nonincreasing noise, with sharp entropy–moment inequalities providing explicit slack terms (Madiman et al., 2018).

7. Synthesis, Significance, and Future Directions

Entropy–capacity logprob-native inference constitutes a unifying intellectual thread across rational updating, information theory, generative model deployment, and dynamic computation. It provides a principled basis for:

• Quantifying and controlling uncertainty relative to available information resources.
• Designing algorithms that match modeling capacity to inferential difficulty in real time.
• Formalizing the boundaries for safe deployment of generative models in high-stakes and resource-limited settings.
• Establishing foundations for objective prior selection and identification of overfitting via rigorous capacity metrics.

Open questions remain in fully generalizing these principles to hierarchical, non-i.i.d., or singular model classes, and in operationalizing objective indifference principles in modern deep learning contexts (LaMont et al., 2017, Caticha, 2021, Singha, 2 Dec 2025, Simonds, 5 Feb 2025, Madiman et al., 2018). Empirically, validation across broader domains and under naturally occurring (rather than synthetic) inputs is a frontier for future research.