Intersection property and interaction decomposition

The decomposition of interactions, or interaction decomposition, is a hierarchical decomposition of the factor spaces into direct sums of interaction spaces. We consider general arrangements of vector subspaces and want to know when one can still build a similar decomposition which one can see as a generalised Gram-Schmidt process. We show that they are exactly the ones that verify an intersection property of the same type as the Bayesian intersection property. By doing so we want to put in perspective the significance of the intersection property. In this presentation we focus on the statement of the intersection property, practical equivalent statements for well-founded posets, the notion of decomposability and the main equivalence theorems between the intersection property and decomposability, leaving for an other redaction the many, old and new, examples in which this decomposition comes in handy. An application of the equivalence theorem, is a interaction decomposition for factor spaces of any collection of random variables which extends the usual one.


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