Atomic Intersection of s-Fields and Some of Its Consequences - Patrizia Berti; Luca Pratelli; Pietro Rigo (Università degli Studi di Pavia, Dipartimento di Economia Politica e Metodi Quantitativi (EPMQ), 2009)
Let (omega,F,P) be a probability space. For each G in F, define G as the s-field generated by G and those sets f in F satisfying P(f) in 0, 1. Conditions for P to be atomic on the intersection of the complements of Ai for i=1,..,k, with A1, . . . ,Ak in F sub-s-fields, are given. Conditions for P to be 0-1-valued on the intersection of the complements of Ai for i=1,..,k are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.
Keywords: 330; Atomic probability measure; Gibbs sampling; Graphical models; Intersection property; Iterated conditional expectations