Factorization of the Cumulative Distribution Function in Case of Conditional Independence. (REVISED, November 1999)

A decomposition of complex estimation problems is often obtained by
using factorization formulas for the underlying likelihood or
density function. This is, for instance, the case in so-called
decomposable graphical models where under the restrictions of
conditional independences induced by the graph the estimation in
the original model may be decomposed into estimation problems
corresponding to subgraphs. Such a decomposition is based on the
property of conditional independence which can be read off the
graph and on the factorization of the assumed underlying density
function. In this paper analogous factorization formulas for the
cumulative distribution function are introduced which can be useful
in situations where the density is not tractable.