Bayesian inference
In plain wordsReasoning 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.
Statistical inference in which unknown quantities — parameters, hypotheses, latent variables, future observations — are given probability distributionsprobability 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. representing uncertainty, and conclusions follow by conditioning on data with 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.: a 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. combined with 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. gives a posteriorposterior, n.: What you believe after weighing new evidence against what you believed before. It is the updated belief that Bayes' theorem produces., and predictions average over the posterior. Exact posteriors are rare, so practice relies on conjugate models, Markov chain Monte CarloMarkov chain Monte Carlo, n.: A family of methods for drawing random examples from a complicated probability distribution by taking a long, carefully designed random walk that eventually visits each region in proportion to its probability., 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., Laplace approximations and sequential Monte Carlo. In machine learning the Bayesian approach underlies Gaussian processesGaussian process, n.: A way of predicting an unknown curve or surface that gives not just a best guess at each point but an honest error bar, narrow near measured points and wide far from them., 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., 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., Thompson samplingThompson sampling, n.: A way to choose among uncertain options by picking each one about as often as it is likely to be the best, based on everything learned so far., much of 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., and accounts 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. as implicit Bayesian inference. It contrasts with frequentist statistics, which treats parameters as fixed and probabilities as long-run frequencies; debates between the schools shaped 20th-century statistics, and modern practice often mixes them.
Bayesian inference gives a posterior over the conversion rate, so we can say how likely variant B is to be better.
Origin
Thomas Bayes (1763) and especially Pierre-Simon Laplace developed the approach, long called 'inverse probability'. Ronald Fisher attacked it, and the adjective 'Bayesian' entered circulation around 1950, appearing in Fisher's Contributions to Mathematical Statistics (1950). Revived by Harold Jeffreys, Leonard Savage and Bruno de Finetti, it became computationally practical with MCMC methods around 1990.
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.
- 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.
- posteriorposterior, n.: What you believe after weighing new evidence against what you believed before. It is the updated belief that Bayes' theorem produces.
- 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.
- Markov chain Monte CarloMarkov chain Monte Carlo, n.: A family of methods for drawing random examples from a complicated probability distribution by taking a long, carefully designed random walk that eventually visits each region in proportion to its probability.
- 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.
- 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.
- maximum likelihood estimationmaximum likelihood estimation, n.: Tuning a model's settings until the real data it learns from looks as likely, and so as unsurprising, to the model as possible. Most AI training, including the pretraining of chatbots, is a version of this.
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., 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., categorical distributioncategorical distribution, n.: The probabilities for a single pick among a fixed set of options, like a weighted die. A language model's guess about the next word is one of these, spread over its whole vocabulary., fuzzy logicfuzzy logic, n.: A kind of logic that allows degrees of truth, so a statement like 'the room is warm' can be 70 percent true, which helps machines handle vague everyday categories., Gaussian processGaussian process, n.: A way of predicting an unknown curve or surface that gives not just a best guess at each point but an honest error bar, narrow near measured points and wide far from them., I. J. GoodI. J. Good, prop. n.: A British statistician who broke codes with Alan Turing and in 1965 argued that a machine smarter than us could design still smarter machines, setting off an 'intelligence explosion'., inductive reasoninginductive reasoning, n.: Reasoning from particular examples to a general rule or prediction — every swan I have seen is white, so swans are probably white — which is useful but can always turn out wrong., inferenceinference, n.: Drawing a conclusion from evidence. In today's AI it most often means running an already-trained model to get an answer, as opposed to training it., Judea PearlJudea Pearl, prop. n.: An Israeli-American computer scientist who gave AI the mathematics for reasoning under uncertainty and, later, for telling cause and effect apart from mere correlation., Kalman filterKalman 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., latent Dirichlet allocationlatent Dirichlet allocation, n.: A popular topic-finding method that assumes every document is a blend of a few themes and every theme favors certain words, then works backward from the words to find both., 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., maximum likelihood estimationmaximum likelihood estimation, n.: Tuning a model's settings until the real data it learns from looks as likely, and so as unsurprising, to the model as possible. Most AI training, including the pretraining of chatbots, is a version of this., posteriorposterior, n.: What you believe after weighing new evidence against what you believed before. It is the updated belief that Bayes' theorem produces., predictive processingpredictive processing, n.: The idea that the brain is a prediction machine: it constantly guesses what its senses will report next and learns mainly from its mistakes, so what we perceive is largely its best guess., 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., p-valuep-value, n.: A number saying how surprising your data would be if nothing interesting were going on. A small p-value hints at a real effect, but it is not the probability that the effect is real., 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., Bayesian information criterionBayesian information criterion, n.: A score for comparing statistical models that, like AIC, rewards good fit and penalizes extra adjustable numbers, but penalizes them more as the dataset grows, favoring simpler models. Lower is better., Bayesian neural networkBayesian neural network, n.: A neural network that keeps a whole range of plausible settings for its internal numbers instead of one best guess, so that it can say how unsure it is about each prediction., 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., Dempster–Shafer theoryDempster–Shafer theory, n.: A theory of reasoning with uncertain evidence that lets you hold back belief — in effect saying 'I don't know' — instead of being forced to divide all your confidence among the possibilities., elastic weight consolidationelastic weight consolidation, n.: A way to help a neural network learn a new task without forgetting an old one, by making the settings that mattered most for the old task harder to change, as if tied down with springs.