Summary  
This chapter covers computing the entropy of a probability distribution to measure uncertainty in next-token predictions and shows how transformer components (embeddings, self-attention, MLP blocks) iteratively reduce that entropy to yield more confident outputs.

General domain of usage  
Natural language generation

Understanding how large language models (LLMs) process and generate text requires a grasp of key ideas from information theory. One of the most fundamental concepts is **entropy**. In information theory, entropy measures the average amount of uncertainty or surprise associated with predicting the next token in a sequence. When you use a language model to generate text, it assigns probabilities to each possible next token. If these probabilities are spread out—meaning the model is unsure — **entropy is high**. If the model is confident and one token has a much higher probability than the others, **entropy is low**.

In the context of language modeling, entropy quantifies how predictable or unpredictable the next token is. A lower entropy means the model finds the text sequence more predictable, while higher entropy suggests greater uncertainty. This is crucial for understanding both how LLMs learn language patterns and how they generate coherent, human-like text. The goal during training is often to minimize entropy by improving the model’s ability to predict the next token accurately, which translates to more fluent and contextually appropriate outputs.

**Compression** in information theory refers to representing data using as few bits as possible without losing essential information. In the context of LLMs, compression is about encoding language patterns efficiently so that meaningful information (**signal**) is preserved while redundant or irrelevant parts (**noise**) are minimized. The distinction between signal and noise in LLM outputs is important: **signal** refers to the meaningful, contextually appropriate content, while **noise** includes randomness, errors, or irrelevant details that do not contribute to the intended message.

Definition

No matter how advanced an LLM is, there are fundamental limits to how well it can predict the next token in a sequence. These limits arise from the inherent unpredictability of natural language—due to ambiguity, creativity, and the presence of truly random or novel information in text. Even with perfect training data, some uncertainty always remains because language is not a closed or fully deterministic system. This unpredictability is reflected in the **entropy** of the model’s output distribution: there will always be some irreducible uncertainty, especially in open-ended or creative tasks.

Limits of Predictability in Language Generation

Because of these limits, LLMs will sometimes generate unexpected or "hallucinated" content, especially in cases where the training data does not provide a clear answer or where multiple plausible continuations exist. Understanding these boundaries helps set realistic expectations for LLM performance and guides the development of strategies to handle uncertainty, such as adjusting sampling methods or using external knowledge sources.

Practical Implications for LLMs

In a transformer-based LLM, information flows through several stages that progressively reduce uncertainty about the next token. The input tokens are first mapped into embeddings, which provide an initial, context-free representation. Self-attention layers then integrate contextual information from across the sequence, refining the model’s understanding of relationships and dependencies. MLP blocks further transform these representations, shaping them into richer semantic features. By the time the model produces its output distribution, most of the initial uncertainty has been resolved. A sharp output distribution indicates low entropy and high confidence, while a flatter distribution reflects higher uncertainty about which token should come next.


What does entropy represent in the context of large language models (LLMs)?

A comprehensive, code-free exploration of transformer-based language models, focusing on their architecture, text generation mechanics, and the theoretical principles underlying their behavior.

Explore the essential components that enable transformers to process and represent language, including attention, residual pathways, and positional encodings.

Examine the theoretical underpinnings of text generation in transformers, from probability distributions to sampling strategies.

Dive into optimization, error modes, and the information-theoretic perspective on transformer-based language models.

Information-Theoretic View of LLMs

Information-Theoretic View of LLMs