conditional probability
In plain wordsThe chance of something once you already know that something else is true — for example, the chance of rain given that the sky is cloudy.
The probability of an event A given that event B has occurred, written P(A | B) and defined as P(A and B) divided by P(B) when P(B) > 0; conditioning keeps only the outcomes compatible with what is known and renormalizes. Two events are independent when conditioning on one leaves the other's probability unchanged; conditional independence — independence given a third variable — is the structural assumption behind naive Bayesnaive Bayes, n.: A simple, fast way of sorting items into categories using probabilities, which pretends that each clue is independent of the others. It is the classic technique behind early email spam filters., hidden Markov modelshidden Markov model, n.: A model of a sequence, such as speech or DNA, in which hidden states — say, the words being spoken — change step by step at random, and each state gives off signals that can be observed, such as sounds. and Bayesian networksBayesian network, n.: A diagram of causes and effects with probabilities attached — say, how rain and a sprinkler can each make grass wet — that lets a computer update its beliefs about hidden causes when it sees evidence.. Conditional probability is the core of modern AI: Bayes' theoremBayes' theorem, n.: A formula for updating how much you believe something after seeing new evidence: weigh your earlier belief by how well that explanation predicts what you actually observed. inverts it to update beliefs, a classifier models P(label | input), and a language modellarge language model, n.: A computer program trained on enormous amounts of text to predict the words that come next. At vast scale, that simple skill lets it chat, answer questions, translate, summarize and write code. models P(next token | preceding tokens), from which the chain rulechain rule, n.: A rule for working out how a change ripples through a chain of steps by combining the effect at each step. It is how AI training traces each error back through every layer. of probability builds probabilities of whole sequences. Confusing P(A | B) with P(B | A), the 'prosecutor's fallacy', is a frequent error in reasoning about tests and detectors.
Conditional on the prompt containing a code block, the model answered in Python 94 percent of the time.
Origin
Abraham de Moivre's Doctrine of Chances (1718; 2nd ed. 1738) treated dependent and independent events and the multiplication rule, and Thomas Bayes's essay of 1763 reasoned about conditioning on observations. Andrey Kolmogorov (1933) made the ratio the definition and developed conditioning on random variables in measure-theoretic form.
Related
- Bayes' theoremBayes' theorem, n.: A formula for updating how much you believe something after seeing new evidence: weigh your earlier belief by how well that explanation predicts what you actually observed.
- chain rulechain rule, n.: A rule for working out how a change ripples through a chain of steps by combining the effect at each step. It is how AI training traces each error back through every layer.
- 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.
- naive Bayesnaive Bayes, n.: A simple, fast way of sorting items into categories using probabilities, which pretends that each clue is independent of the others. It is the classic technique behind early email spam filters.
- Bayesian networkBayesian network, n.: A diagram of causes and effects with probabilities attached — say, how rain and a sprinkler can each make grass wet — that lets a computer update its beliefs about hidden causes when it sees evidence.
- autoregressive modelautoregressive model, n.: A model that produces a sequence one piece at a time, each new piece depending on the pieces already produced — like writing a sentence word by word without going back.
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., Bayes' theoremBayes' theorem, n.: A formula for updating how much you believe something after seeing new evidence: weigh your earlier belief by how well that explanation predicts what you actually observed., chain rulechain rule, n.: A rule for working out how a change ripples through a chain of steps by combining the effect at each step. It is how AI training traces each error back through every layer., 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., do-calculusdo-calculus, n.: A set of mathematical rules, devised by Judea Pearl, for working out the effect of deliberately changing something using only observational data plus a diagram of what causes what.