Bayes' theorem

The identity relating the two conditional probabilitiesconditional probability, n.: The chance of something once you already know that something else is true — for example, the chance of rain given that the sky is cloudy. P(H | E) and P(E | H): the posteriorposterior, n.: What you believe after weighing new evidence against what you believed before. It is the updated belief that Bayes' theorem produces. probability of a hypothesis H given evidence E equals the likelihoodlikelihood, n.: A score for a possible explanation of some data: how probable that explanation, or that setting of a model, would have made the data actually seen. of the evidence under H times the priorprior, n.: What you believe, and how strongly, before looking at the evidence in front of you. In AI it also means built-in assumptions that steer what a system learns. probability of H, divided by the overall probability of the evidence. That denominator, the marginal likelihood or evidence, sums likelihood × prior over all hypotheses and is often intractable. As mathematics the theorem follows directly from the definition of conditional probability and is uncontroversial; what has been debated is using it to assign probabilities to hypotheses and parameters, the defining move of Bayesian inferenceBayesian inference, n.: Reasoning that treats unknowns as uncertain beliefs and updates them with evidence using Bayes' theorem. The result is not one answer but a range of answers, each with a probability.. In AI it underlies 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. classifiers and spam filters, 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., KalmanKalman filter, n.: A method that keeps a running best guess of something you cannot measure exactly, such as a vehicle's position, by blending predictions from a motion model with noisy sensor readings. and particle filtersparticle filter, n.: A way to track something uncertain, such as a robot's location, by keeping thousands of random guesses, scoring each against what the sensors see, and copying the best ones., Bayesian optimizationBayesian optimization, n.: A smart way to tune settings when each test is expensive: build a statistical guess of how settings affect results, then try whichever setting that guess says is most promising next. and Bayesian readings of in-context learningin-context learning, n.: A language model's ability to pick up a new task from instructions or examples placed in its input, on the spot, without any retraining — like a quick study learning a game's rules from one round.. Ignoring the prior produces the base-rate fallacy.

With a base rate of 1 in 1,000, Bayes' theorem shows that most alarms from a 99-percent-accurate detector are false.