Entropy in Large Language Models

Evaluating whether large language models (LLMs) capture the structure of natural language beyond local fluency remains an open challenge. Existing evaluation methods, largely based on task performance or short-context behavior, provide limited insight into the long-range statistical organization of generated text. We propose a complementary evaluation framework based on repeated subsequences. By analyzing their distribution across scales and relating it to higher-order Rényi entropies, we probe how texts reuse previously established structure under finite-length conditions. Experiments on human-written texts and length-matched GPT-generated texts show that, while power-law models can describe restricted ranges of block length, the observed entropy growth is often equally or better characterized by logarithmic--power forms. Across datasets, natural language exhibits stable entropy-growth patterns over accessible ranges, with consistent average behavior despite variability across individual texts. In contrast, GPT-generated texts show systematic and statistically significant shifts in estimated exponents with model size. These results demonstrate that repeated-subsequence entropy provides a quantitative structural diagnostic that reveals systematic differences in long-range organization, distinguishing natural language from state-of-the-art LLM outputs beyond surface-level fluency.

Humans and large language models can predict next letter or word from its prior context much better than random guessing, indicating strong redundancy of language viewed as a stochastic process. Quantitatively this redundancy was estimated by Shannon to be around 80%, which means that every letter of a printed English text conveys approximately 1 bit of information and not 4.8 bits that 27 letters (including spaces) could potentially carry. This estimate was later confirmed by using autoregressive token probabilies computed by large language models. However, the statistical organization of language that give rise to such a large redundancy remains unclear. Here we introduce a statistical framework of language linking its redundancy to the hierarchical semantic organization of text. To this end, we use large language models to recursively segment any given text into semantically coherent chunks, inducing a semantic tree'' that spans the whole range of text organization, beginning from its main idea to individual tokens (words). For a large corpus of texts of a particular type, say fiction stories, the resulting ensemble of semantic trees is characterized by specific statistical regularities, giving rise to a structural'' entropy rate defined in this study. Surprisingly, we discovered that for several datasets considered in this work, semantic tree entropy rate was quite close to LLM-measured quantity and exhibited a similar trend across corpus. In particular, simpler texts like children stories exhibit lower branching in their semantic trees and correspondingly lower entropy rates, whereas fiction and poetry exhibit progressively larger branching factors and greater entropy rates. These results suggest that hierarchical semantic organization of texts is an important factor in their overall information transmission rates.

This paper presents a probabilistic perspective on large language models (LLMs), developed with the aim of bringing together, in a single self-contained account, tools that are usually treated separately across the literature. LLMs are described through probability measures on the set of sequences of tokens, specified via their autoregressive conditional distributions. Training is formulated as a maximum-likelihood estimation problem, addressed by stochastic gradient methods, while text generation is viewed as the sequential simulation of the resulting stochastic process. The role of the asymmetry of the Kullback--Leibler divergence in text generation is examined in relation with characteristic phenomena such as hallucination and the distinction between statistical plausibility and truth. As a complementary illustration of the same viewpoint, we also discuss diffusion models, built around the score function, which cast generation not as sequential token prediction but as the simulation of a reverse-time stochastic process transforming noise into data both in discrete and continuous time.