Mathematics and statistics

Exact sequence

An exact sequence is a sequence of algebraic objects, typically modules, connected by homomorphisms in which each map's image equals the next map's kernel, the central notion of homological algebra.

Exact test

An exact test is a statistical hypothesis test whose Type I error rate stays at or below the significance level in finite samples, including Fisher's exact test.

Exchange matrix

The exchange matrix J is the square matrix with ones on the antidiagonal and zeros elsewhere; multiplying by it reverses a matrix's rows or columns.

Exchangeability, known in older literature as equivalence, permutability, or symmetry, is a symmetry property of a probability distribution, introduced by Bruno de Finetti and central to Bayesian statistics.

Exchangeable random variables

Exchangeable random variables are a collection whose joint probability distribution is unchanged when their labels are permuted, introduced by Bruno de Finetti and weaker than independence.

Exclusion process

An exclusion process is a stochastic interacting particle system where particles hop in continuous time on discrete sites, at most one per site; its asymmetric form, the ASEP, is a paradigm of non-equilibrium statistical physics.

Exclusive or

Exclusive or (XOR, exclusive disjunction) is the logical connective that yields true when exactly one of its two operands is true, written ⊕ or ⊻.

Existential quantification

Existential quantification, or the existential quantifier, is the logical operation written ∃ asserting a formula holds for at least one value of a variable, one of the two standard quantifiers of first-order logic.

Existentially closed model

An existentially closed model, also called an existentially complete model, is a model of a theory that already solves every existential condition solvable in an extension.

Exotic R4

An exotic R4 is a smooth manifold homeomorphic to ordinary Euclidean four-space but not diffeomorphic to it; dimension four is the only one with such structures, and uncountably many exist.

Exotic sphere

An exotic sphere is a smooth manifold that is topologically a sphere but has a different smooth structure, first discovered by John Milnor in dimension 7 in 1956.

Expander graph

An expander graph is a sparse but highly connected graph, usually studied as constant-degree families, with applications in network design, error-correcting codes, and proof systems.

Expectation propagation

Expectation propagation (EP) is a deterministic approximate Bayesian inference algorithm, introduced by Minka in 2001, that approximates an intractable posterior by iteratively matching moments of local site approximations.

Expectation–maximization algorithm

The expectation–maximization (EM) algorithm, formalized by Dempster, Laird, and Rubin in 1977, is an iterative method for maximum likelihood estimation in statistical models with unobserved latent variables or incomplete data.

Expectation, moments and probability inequalities

Expectation, moments and probability inequalities is an area of probability theory in which expectation and moments measure random variables and inequalities such as Markov's and Chebyshev's bound their deviations.

Expected information gain

Expected information gain is a criterion in Bayesian experimental design measuring how much an experiment reduces uncertainty about unknown parameters; pioneered by Dennis Lindley in 1956.

Expected utility hypothesis

The expected utility hypothesis, also called expected utility theory, says a rational decision maker chooses among risky options by picking the act with the highest expected utility.

Expected value

The expected value, also called the mathematical expectation or mean, is the probability-weighted average of a random variable over repeated trials. Blaise Pascal introduced the concept in 1654.

Expected value of sample information

The expected value of sample information (EVSI) is a decision-theoretic measure of the average worth of a proposed study, the expected gain from updating a decision with its data.

Expectile regression

Expectile regression estimates conditional expectiles, asymmetric least-squares analogs of quantiles, using smooth squared loss that is computationally simpler than quantile regression and carries tail magnitude information.

Explicit formulae (L-function)

An explicit formula for an L-function is an identity in analytic number theory expressing a sum over the L-function's nontrivial zeros as a sum over prime powers.

Explicit time integration

Explicit time integration is a class of numerical methods that advance time-dependent differential equations step by step using only known values, favored for crash and shock simulation.

Exploratory causal analysis

Exploratory causal analysis, also known as causal discovery or data causality, uses observational data to learn the causal structure of a system of variables, typically as a causal graph.

Exploratory data analysis

Exploratory data analysis, also called explorative data analysis, is a statistical approach that examines data first with descriptive statistics and graphs before hypothesis testing, named by John Tukey in 1977.

Exploratory factor analysis

Exploratory factor analysis (EFA) is a multivariate statistical method that models correlations among observed variables as the shared influence of a smaller number of unobserved common factors, without imposing a factor structure in advance.

Exploratory graph analysis

Exploratory graph analysis (EGA) is a network psychometrics method that estimates the number of dimensions in data via community detection on a regularized correlation network.

Exploratory structural equation modeling

Exploratory structural equation modeling (ESEM) is a statistical framework, introduced by Tihomir Asparouhov and Bengt Muthén in 2009, that combines exploratory factor analysis with the confirmatory machinery of structural equation modeling.

Exponential distribution

The exponential distribution, also called the negative exponential distribution, is a continuous probability distribution describing waiting times between independent events occurring at a constant average rate.

Exponential family

The exponential family is a class of probability distributions sharing one exponential density form, including the Gaussian, binomial, Poisson, and gamma, and underlying generalized linear models.