expected value
In plain wordsThe 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.
The probability-weighted average of all the values 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. can take — a sum over outcomes for discrete variables, an integral against the densityprobability density function, n.: A curve showing where a measurement is likely to fall. The area under the curve between two values gives the chance of landing between them. for continuous ones — written E[X]. Expectation is linear: E[aX + bY] = aE[X] + bE[Y] even when X and Y are dependent, and the law of large numberslaw of large numbers, n.: The mathematical rule that averages of chance events steady out over many repetitions: flip a fair coin a few times and the share of heads can be far from one-half, but over thousands of flips it gets very close. guarantees that sample averages converge to it. Machine-learning objectives are expectations: training minimizes the expected lossloss function, n.: A score that measures how wrong a model's predictions are. Training means adjusting the model, step by step, to push this number down. over the data distribution, approximated by averages over mini-batches; 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. maximizes expected returnreturn, n.: The total reward an AI agent collects from now until the end of a task, usually counting near-term rewards more than distant ones. Agents are trained to make this total as large as possible.; and variational inferencevariational inference, n.: A way of approximating a hard-to-compute probability distribution by picking the closest match from a simpler family of distributions, turning a hard problem of adding up every possibility into an optimization problem. and the evidence lower boundevidence lower bound, n.: A quantity that is always less than or equal to how well a model explains its data, but much easier to calculate; pushing it upward is how models such as VAEs learn. are written as expectations estimated by Monte CarloMonte Carlo method, n.: Estimating an answer by running many random trials and averaging the results — like estimating the odds in a card game by dealing thousands of hands instead of calculating. sampling. Decision theory recommends maximizing expected utilityutility function, n.: A way of scoring every possible outcome by how much someone, or some AI, wants it. A rational agent is often modeled as choosing whatever maximizes its expected score., and variancevariance, n.: A number that grows the more widely something's possible values scatter away from its average, and is zero if it never varies. Its square root, the standard deviation, uses the same units as the values themselves. measures spread around the expectation.
Because expectation is linear, the expected batch loss is just the average of the per-example expected losses.
Origin
Christiaan Huygens's 1657 treatise on games of chance, published in Frans van Schooten's Latin, reasoned with the fair value of a gamble, building on the 1654 correspondence of Blaise Pascal and Pierre de Fermat. 'Expectation', 'mathematical expectation' and 'expected value' became standard in probability texts of the 18th to 20th centuries, and W. A. Whitworth introduced the symbol E in 1901.
Usage note
'Mean' can denote either the expected value of a distribution or the average of a sample; the sample mean is an estimate of the expected value, not the same thing.
Related
- variancevariance, n.: A number that grows the more widely something's possible values scatter away from its average, and is zero if it never varies. Its square root, the standard deviation, uses the same units as the values themselves.
- 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.
- law of large numberslaw of large numbers, n.: The mathematical rule that averages of chance events steady out over many repetitions: flip a fair coin a few times and the share of heads can be far from one-half, but over thousands of flips it gets very close.
- Monte Carlo methodMonte Carlo method, n.: Estimating an answer by running many random trials and averaging the results — like estimating the odds in a card game by dealing thousands of hands instead of calculating.
- utility functionutility function, n.: A way of scoring every possible outcome by how much someone, or some AI, wants it. A rational agent is often modeled as choosing whatever maximizes its expected score.
- probability distributionprobability distribution, n.: A complete description of how likely each possible outcome is — for a fair die, one-sixth for each face. A language model produces one of these over its whole vocabulary at every step.
Mentioned in
probabilityprobability, n.: How likely something is, on a scale from 0 for the impossible to 1 for the certain; rolling a six with a fair die scores one-sixth. AI systems state their predictions, even each next word, as probabilities., 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., entropyentropy, n.: A 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., 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., law of large numberslaw of large numbers, n.: The mathematical rule that averages of chance events steady out over many repetitions: flip a fair coin a few times and the share of heads can be far from one-half, but over thousands of flips it gets very close., Monte Carlo methodMonte Carlo method, n.: Estimating an answer by running many random trials and averaging the results — like estimating the odds in a card game by dealing thousands of hands instead of calculating., 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., utility functionutility function, n.: A way of scoring every possible outcome by how much someone, or some AI, wants it. A rational agent is often modeled as choosing whatever maximizes its expected score., variancevariance, n.: A number that grows the more widely something's possible values scatter away from its average, and is zero if it never varies. Its square root, the standard deviation, uses the same units as the values themselves., covariancecovariance, n.: A measure of whether two quantities tend to rise and fall together, move in opposite directions, or show no consistent relationship., decision theorydecision theory, n.: The study of how to make the best choice when outcomes are uncertain, by weighing how likely each result is against how much it matters., Fisher informationFisher information, n.: A number, or table of numbers, saying how sharply the probability of the data reacts when a model's settings are nudged; the sharper the reaction, the more precisely the data can pin those settings down., importance samplingimportance sampling, n.: A shortcut for estimating averages: draw samples from a convenient source, then reweight each one to correct for how much more or less likely it would be under the source you actually care about., Pascal's muggingPascal's mugging, n.: A puzzle about decisions: if someone promises an astronomically huge reward or threatens a huge disaster, even a tiny chance they are honest seems to justify paying them, which looks absurd.