Conditional Pro ducts An Alternative Approach to Conditional
Indep endence
M Studen y A P Dawid
Academy of Sciences of Czech Rep and Department of Statistical Science
University of Economics Prague University College London
Pod vod arenskou v e z Prague CZ Gower Street London WC E BT UK
dersto o d as b eing ab out a generalized form of separa Abstract
tion
We introduce a new abstract approach to the
An entirely di erent approach to conditional indep en
study of conditional indep endence founded
dence is to try and abstract the factorization prop
on a concept analogous to the factoriza
erties which form the traditional basis for the proba
tion prop erties of probabilistic indep endence
bilistic de nition This is the task we attempt here
rather than the separation prop erties of a
We rst introduce the abstract concept of conditional
graph The basic ingredient is the con
product and prop ose a suitable axiom system for it
ditional pro duct which provides a way of
We use this to introduce conditional indep endence as a
combining the ob jects under consideration
derived concept and show that it do es then satisfy the
while preserving as much indep endence as
semi graphoid axioms Such a p oint of view provides a
p ossible We introduce an appropriate ax
unifying framework for conditional indep endence and
iom system for conditional pro duct and
suggests new forms and applications However not ev
show how when these axioms are ob eyed
ery semi graphoid can arise in this way In this pap er
it induces a derived concept of conditional
we describ e the appropriate conditional pro duct con
indep endence which ob eys the usual semi
struction in probabilistic and set theoretic frameworks
graphoid axioms The general structure is
for conditional indep endence as well as showing that
used to throw light on three sp eci c ar
in the frameworks of undirected or directed graphical
eas the familiar probabilistic framework
mo dels no suitable conditional pro duct exists