Bayes' theorem

Mathematical rule for inverting probabilitiesIn probability theory, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes (), gives a mathematical rule for inverting conditional probabilities, allowing the probability of a cause to be found given its effect. For example, with Bayes' theorem, the probability that a patient has a disease given that they tested positive for that disease can be found using the probability that the test yields a positive result when the disease is present.

Conditional probabilityProbability of an event occurring, given that another event has already occurredIn probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion, or evidence) is already known to have occurred.

Prior probabilityA prior probability distribution (often simply called the prior probability, prior distribution, or prior) of an uncertain quantity is its assumed probability distribution before evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election.

Bayesian probabilityBayesian probability ( or ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown.Bayesian statisticsBayesian statistics ( BAY-zee-ən or BAY-zhən) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a degree of belief in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event.

Bayesian inferenceMethod of statistical inferenceBayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian inference uses a prior distribution to estimate posterior probabilities. Bayesian inference is an important technique in statistics, and especially in mathematical statistics.

Posterior probabilityThe posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule.

Likelihood functionFunction related to statistics and probability theoryA likelihood function (often simply called the likelihood) gives the relative merit of various statistical models for describing a data set. Often the models being compared are parameterized by a parameter, with the parameter often written as θ, or they are parameterized by multiple parameters given as the components of a vector.

Probability theoryProbability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms.

Realization (probability)In probability and statistics, a realization or observation (also called observed value) of a random variable or random element is the value that is actually observed or measured. For example, if the random variable is human height, a given realization could be "2 meters”. The random variable itself is the process dictating how the observation comes about.

Thomas BayesThomas Bayes ( BAYZ; c. 1701 – 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem. Bayes never published what would become his most famous accomplishment; his notes were edited and published posthumously by Richard Price.

Pierre-Simon LaplaceFrench polymath (1749–1827)Pierre-Simon, Marquis de Laplace (; French:; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique céleste (Celestial Mechanics) (1799–1825).

Statistical inferenceStatistical inference is the process of using data analysis to infer properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics.

*As an Amazon Associate I earn from qualifying purchases.