entropy
In plain wordsA measure of unpredictability. A fair coin toss has more entropy than a rigged one; an AI that is unsure of its next word has high entropy.
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Information theory
The expectedexpected value, n.: The long-run average you would get if a chancy situation could be repeated many times — 3.5 for a fair die, even though no single roll shows 3.5. surprise of a random variablerandom variable, n.: A quantity whose value depends on chance, such as the number rolled on a die or the next word a chatbot will write.: the average over outcomes of −log p(x), measuring how uncertain its distribution is. Entropy is zero for a certain outcome and maximal, log K, for a uniform distribution over K values; for continuous variables the analogue is differential entropy. By Shannon's source coding theorem it is the minimum average number of bits per symbol needed to encode samples losslessly. In machine learning, the entropy of a model's predictive distribution measures its uncertainty and is used to choose examples in active learningactive learning, n.: A way to train a model with fewer labeled examples by letting it choose which unlabeled examples a person should label next, usually the ones it is most unsure about., flag out-of-distributionout-of-distribution, adj.: Describes inputs that differ from anything an AI system saw in training — like a model trained on daytime photos facing a night scene — where its answers become less reliable. inputs and monitor language-model generation; entropy regularizationentropy regularization, n.: Rewarding an AI agent for keeping some randomness in its choices, so it keeps exploring alternatives instead of locking in on one behavior too early. rewards high-entropy policies in reinforcement learningreinforcement learning, n.: A way of teaching a computer program by trial and error: it tries actions, earns points for good results and penalties for bad ones, and gradually learns which choices pay off over time. to sustain exploration, and collapsing policy entropy is a known failure in RL training of language models; information gaininformation gain, n.: How much a question reduces uncertainty about the answer. A decision tree asks first the question that removes the most uncertainty, measured in bits. for decision treedecision tree, n.: A model that makes a prediction by asking a series of yes-or-no questions about the input, like a flowchart, with each answer leading to the next question until it reaches a final answer. splits is a reduction in entropy. cross-entropycross-entropy, n.: A score of how surprised a model's predictions are by what actually happens; lower is better. Language models are trained mainly by driving this number down on real text. and Kullback–Leibler divergenceKullback–Leibler divergence, n.: A score for the cost of believing the wrong odds: on average, how much more surprised you will be by real outcomes than someone who knew the true probabilities. It is zero only when your odds are exactly right. are measured from it.
Token-level entropy spiked whenever the model reached an unfamiliar API name.
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Physics
In thermodynamics and statistical mechanics, a measure of disorder: the logarithm of the number of microscopic arrangements consistent with a macroscopic state (Boltzmann's S = k log W). Shannon's formula has the same form as the Gibbs entropy of statistical mechanics, a link exploited by energy-based modelsenergy-based model, n.: A model that scores how well things fit together by giving each possibility an 'energy': low for plausible combinations, high for implausible ones, like a ball settling into the lowest valleys. and maximum-entropy methods.
Origin
Rudolf Clausius coined 'entropy' for thermodynamics in 1865, from Greek tropē, 'transformation', and Ludwig Boltzmann gave it a statistical interpretation in the 1870s. Claude Shannon adopted the word for his measure of information in 1948 — reportedly on John von Neumann's advice, a much-repeated anecdote that may be apocryphal.
Related
- information theoryinformation theory, n.: A branch of mathematics that treats information as surprise — the less predictable a message, the more it tells you — and works out how far data can be compressed and how reliably it can be sent.
- cross-entropycross-entropy, n.: A score of how surprised a model's predictions are by what actually happens; lower is better. Language models are trained mainly by driving this number down on real text.
- Kullback–Leibler divergenceKullback–Leibler divergence, n.: A score for the cost of believing the wrong odds: on average, how much more surprised you will be by real outcomes than someone who knew the true probabilities. It is zero only when your odds are exactly right.
- mutual informationmutual information, n.: How much knowing one thing tells you about another — for example, how much a word reveals about the topic of a document. It is zero when the two are unrelated.
- entropy regularizationentropy regularization, n.: Rewarding an AI agent for keeping some randomness in its choices, so it keeps exploring alternatives instead of locking in on one behavior too early.
- information gaininformation gain, n.: How much a question reduces uncertainty about the answer. A decision tree asks first the question that removes the most uncertainty, measured in bits.
Mentioned in
Claude ShannonClaude Shannon, prop. n.: An American engineer and mathematician who invented information theory, the mathematics of bits and communication, and wrote one of the first plans for a chess-playing computer., cross-entropycross-entropy, n.: A score of how surprised a model's predictions are by what actually happens; lower is better. Language models are trained mainly by driving this number down on real text., Gini impurityGini impurity, n.: A score for how mixed a group is: zero when every item belongs to the same category, higher when categories are jumbled together. Decision trees choose splits that make groups purer., information gaininformation gain, n.: How much a question reduces uncertainty about the answer. A decision tree asks first the question that removes the most uncertainty, measured in bits., information theoryinformation theory, n.: A branch of mathematics that treats information as surprise — the less predictable a message, the more it tells you — and works out how far data can be compressed and how reliably it can be sent., Kullback–Leibler divergenceKullback–Leibler divergence, n.: A score for the cost of believing the wrong odds: on average, how much more surprised you will be by real outcomes than someone who knew the true probabilities. It is zero only when your odds are exactly right., normal distributionnormal distribution, n.: The familiar bell-shaped curve: most values cluster near the average, and values become rapidly rarer the farther they lie from it. Measurement errors and adult heights roughly follow it., entropy collapseentropy collapse, n.: A failure in reinforcement-learning training where an AI quickly becomes overconfident and stops trying different answers, so it explores less and its improvement stalls., entropy regularizationentropy regularization, n.: Rewarding an AI agent for keeping some randomness in its choices, so it keeps exploring alternatives instead of locking in on one behavior too early., ID3ID3, abbr.: An early, influential method for automatically building a decision tree from examples, choosing at each step the question that most reduces uncertainty about the answer., irreducible lossirreducible loss, n.: The part of a model's error that no amount of extra size, data or training can remove, because the thing being predicted is itself partly random — like the unpredictability of the next word in an ordinary sentence., Jensen–Shannon divergenceJensen–Shannon divergence, n.: A balanced, two-way measure of how different two sets of probabilities are, comparing each with their average. Unlike its better-known cousin, it gives the same answer in both directions., Kolmogorov complexityKolmogorov complexity, n.: How briefly something can be described by a computer program that recreates it exactly: a million zeros needs only a tiny program, while a truly random string needs one about as long as itself., mutual informationmutual information, n.: How much knowing one thing tells you about another — for example, how much a word reveals about the topic of a document. It is zero when the two are unrelated., semantic entropysemantic entropy, n.: A way to tell when an AI is likely making something up: ask it the same question several times and check whether the answers mean the same thing. Answers that keep changing signal guessing., uncertainty quantificationuncertainty quantification, n.: Measuring how unsure an AI system's predictions are, and why, so that people know when to trust an answer and when to double-check it.