Separateindependent

Separateindependent is a term used in probability theory and statistics to describe a form of independence that respects a predefined partition of a set of random variables into separate groups, or blocks.

Definition: Let I be an index set partitioned into blocks B1, B2, ..., Bm. The family of random variables {X_i : i in I} is called separateindependent if for every choice of finite subcollections corresponding to a selection of blocks, the collections are independent as groups. Equivalently, the sigma-algebras generated by the blocks are independent. Within a block, variables may exhibit arbitrary dependence.

Properties: Separateindependence guarantees independence between the blocks as whole groups, but does not impose independence among variables within a single block. If each block contains exactly one variable, separateindependence reduces to the standard notion of mutual independence for the entire family. It is weaker than full independence unless blocks are singletons, but it provides a modular view of independence aligned with a predefined data or model structure.

Examples: Consider three variables X1, X2, X3 with X1 and X2 potentially dependent, and X3 independent of both. If the partition is {X1, X2} and {X3}, the family {X1, X2, X3} is separateindependent because the block {X1, X2} is independent of the block {X3}, even though X1 and X2 may be dependent.

Applications: The concept appears in modular or block-structured statistical modeling, hierarchical Bayesian networks, and privacy-preserving data analysis where independence is considered at the level of blocks rather than individual variables.

Notes: The term is not widely standardized; some authors refer to similar ideas as block independence or partitioned independence. Interpretations may vary by context.

See also: independence, mutual independence, pairwise independence, block models, modular Bayesian networks.