On Multivariate Structures and Exhaustive Reductions

Simplified representations of multivariate laws, and in particular those allowing one to decrease the dimension while preserving structural information, are of paramount importance in statistical analysis. This paper concerns the theoretical premises of simplification. We introduce a framework that allows us to specify definitions of structure for multivariate laws. Conceiving definitions as partitions of the probability laws on a Euclidean space, we show how they can be generated via partial orders, or binary operations and noise classes. Moreover, the framework allows us to identify simplified representations that are guaranteed to be exhaustive with respect to such definitions, and might live in lower dimension.