Bayesian inference

Bayesian inference updates probability distributions for hypotheses or parameters by combining prior distributions with observed evidence through Bayes’ theorem.

Bayes’ theorem: A mathematical rule relating conditional probabilities, used to update beliefs after observing evidence. It supplies the equation that turns prior probabilities and likelihoods into posterior probabilities.
Conditional probability: The probability of an event given that another event has occurred. Bayes’ theorem reverses conditioning to update hypotheses from observed evidence.
Bayesian statistics: A statistical framework that represents uncertainty with probability distributions and updates them using Bayes’ theorem. It applies Bayesian inference throughout estimation, testing, and prediction.
Frequentist inference: A framework that evaluates procedures through their long-run behavior under repeated sampling. It treats unknown parameters as fixed rather than assigning them prior probability distributions.
Prior probability: A probability distribution representing uncertainty about a hypothesis or parameter before considering current evidence. It encodes the starting uncertainty that the update revises.
Probability distribution: A mathematical description assigning probabilities across possible outcomes or parameter values. Bayesian inference updates distributions rather than only single point estimates.
Bayesian optimization: A strategy for optimizing expensive black-box functions using probabilistic models and sequential evaluations. It uses posterior uncertainty to choose informative, promising experiments.
Maximum likelihood estimation: A method that chooses parameter values maximizing the probability of the observed data. It uses the likelihood alone, unlike Bayesian estimation’s combination of likelihood and prior.
Likelihood function: A function describing how probable observed data are under different parameter values or hypotheses. It measures how well each candidate explanation predicts the observations.
Random variable: A quantity whose possible values are associated with probabilities. Hypotheses, observations, and unknown parameters are represented as random variables.

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