Joindependency

Joindependency is the property that a collection of random variables are independent of one another in the joint sense. In probability theory, a set of random variables is said to be jointly independent if the occurrence of events defined by any combination of the variables does not affect the probabilities of others. The term is often written as joint independence in standard texts, but some sources use the single word form joindependency.

Formally, let X1, X2, ..., Xn be random variables on a common probability space. They are jointly independent if, for every choice of Borel sets B1, B2, ..., Bn in the real line, the probability of the simultaneous events X1 ∈ B1, X2 ∈ B2, ..., Xn ∈ Bn equals the product of the individual probabilities: P(X1 ∈ B1, X2 ∈ B2, ..., Xn ∈ Bn) = ∏i P(Xi ∈ Bi). Equivalently, the joint distribution μ of (X1, ..., Xn) factors as the product of the marginals: μ = μ1 × μ2 × ... × μn. This is also expressed by saying that the variables are independent and their joint behavior is completely determined by their marginals.

Notes and nuances: Joint independence implies pairwise independence, but the converse is not true in general. Independence can be extended to finite collections without requiring independence for all infinite subfamilies, though care is needed in infinite settings. In discrete cases, joint independence means the joint probability mass function factors into the product of the individual pmfs. In continuous cases, the joint probability density function factors into the product of the marginal densities.

Applications and relevance: Independence is a foundational assumption in many statistical methods and probabilistic models. It underpins simple likelihood calculations, Bayesian networks, and algorithms like Naive Bayes. However, real-world data often exhibit dependencies, and testing for joint independence can be nontrivial.