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 vodarenskou vez Prague CZ Gower Street London WCE 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 dierent 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 denition 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

semigraphoid 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 semigraphoid 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 settheoretic frameworks

graphoid axioms The general structure is

for conditional indep endence as well as showing that

used to throw light on three sp ecic ar

in the frameworks of undirected or directed graphical

eas the familiar probabilistic framework

mo dels no suitable conditional pro duct exists

b oth the discrete and the general case a

settheoretic framework related to variation

indep endence and a variety of graphical

CONDITIONAL PRODUCTS

frameworks

In this Section we give an axiomatic denition of pro

jection and conditional pro duct Then we show that

Key words Directed graph indep endence Probabilis

an induced concept of conditional indep endence satis

tic indep endence Pro jection Semigraphoid Undi

es the semigraphoid axioms

rected graph indep endence Variation indep endence

AXIOMS FOR PROJECTION AND

MOTIVATION

CONDITIONAL PRODUCT

In several distinct mathematical areas esp ecially those

We have an index set N a class N of subsets of N

describing uncertainty in and

containing and closed under union and intersection

Dawid and Articial Intelligence Studen y

and a set of ob jects Asso ciated with any

Shenoy some concept of conditional in

is its domain d N

dependence plays a fundamental role p ermitting the

decomp osition of a complex ob ject into simpler pieces

Pro jection

This concept was introduced in miscellaneous frame

works but certain reasonable formal prop erties are

For any and E N there exists an ob ject

E shared These formal prop erties are describ ed by the

the projection of onto E Pro jection has

abstract theory of conditional indep endence as cap

the following prop erties

tured by the semigraphoid axiom system Dawid

E

Sp ohn Pearl The archetypal mo del

P d d E for E N

for this system is the concept of separation in an undi

F

E E F

P for E F N rected graph and the axioms can most readily b e un

d() d( ) d()

Observation and P for

whenever D

F E

E F

The prop erties ab ove imply that and

E d()E

Pro of Use T the fact that C and T to

for E F N In particular

d() d() d( )

E

write Use T

d E implies by P

for the other equality

We say that are weakly compatible if

d( ) d()

Thus when all the ab ove axioms hold and D C we

and strongly compatible if there exists

d() d( )

shall have C C since then we can take

such that and Strong

compatibility implies weak compatibility but not nec

essarily conversely The collection of pairs of weakly

SEMIGRAPHOIDS AND

compatible ob jects from will b e denoted by C and

CONDITIONAL INDEPENDENCE

the collection of pairs of strongly compatible ob jects

from by C

SemiGraphoid

A ternary op eration j on N is called a full

Conditional Pro duct

semigraphoid if it satises

We consider a mapping D having domain

C AB j C if B C

D C often we shall have D C The conditional

pro duct op eration is required to have the following

C AB j C B A j C

prop erties

C AB D j C AD j C

C AB D j C AB j C D

T D d d d

C AB j C D and AD j C AB D j C

T D D and

E E

T E N D and

A partial semigraphoid on a class of triplets K

E

T D d E N D and

N N N is a predicate having K as domain

E

E

such that CC hold under the additional constraint

that all triplets involved b elong to K We shall here

limit attention to sp ecial partial semigraphoids where

Axiom T can b e formulated in apparently stronger

K is the class of triplets of pairwise disjoint nite sub

form

sets of N equivalently CC are only required when

E

A B C D are pairwise disjoint These semigraphoids

T D d d E N D

d()E

E

will b e called disjoint semigraphoids over N

and

Conditional Indep endence

Indeed if d d E one has by P and P

d()E d( )(d()E ) d( )E E

For every and pairwise disjoint nite sets

AB C

A B C N we write AB j C if

E

T D d d E N

AC B C

this includes the requirement

E E

D D and

AC B C

D

E

T d d E N D

E

D D

Observation AB j C i AB j C for

AB C AB C

and

Using T we can also restate T as

Pro of A consequence of P and the denition

T D d d E N

d()E ) d()E )

D and

AC B C

Observation AB j C whenever

AC B C

and D

We can further reexpress TT in terms of pairwise

AC B C

Pro of Put and

E d()E

disjoint A B C D N using by P

AB C

Since d A B C use P and T

AC C C

By P and one can write using T

T D d A C d B C D

AC AC C

and T Similarly

C D AC D C D

D and

B C

using T

T D d A C d B C D

AC D

D and

Prop osition For the collection of triplets

AC D

hA B jC i of pairwise disjoint nite subsets of N for

C D

T d A C d B C D D

which AB j C forms a disjoint semigraphoid

C D

and D D

over N

q on X p q is dened on X as follows Pro of Without loss of generality supp ose b elow that

F E F

the sets A B C D in CC are subsets of d oth

p(xy)q (yz)

E F

if p y

E \F

p (y)

erwise replace every set by its intersection with d

p q x y z

otherwise

With B and C disjoint C b ecomes Aj C

AC

which follows from T with E C note

Q Q

C C

X X y X z for every x

that by P C follows from T with

i i E F

iF nE iE nF

AC B C

From this p oint on dene

Of course if E n F then x is omitted if F n E

AC B C D

E F

If AB D j C then

then z is omitted Of course p in can b e

AB C D AC D

E F

So from P

replaced by q

AC D C D AC C D

by T again

The reader can verify directly

using P so that C holds Also C then follows

easily from T and P Finally AB j C D

Prop osition The properties PP and TT

AB C D C D

and AD j C imply

hold for the above dened projection and conditional

which by T and T with E C D so

product of discrete probability distributions

that C holds

Thus by Prop osition every

p over D N induces a disjoint semigraphoid over

Our axioms for pro jection and conditional pro duct re

N as follows for every triplet hA B jC i of pairwise

semble those for marginalization and combination of

disjoint nite subsets of N one has AB j C p i

valuations Shenoy and Shafer Shenoy

AB C C AC B C

which were motivated by the desire to establish an ax

p x y z p y p x y p y z

iomatic framework for b elief propagation in join trees

Q Q

X X y X z for every x

i i C D

rather than as a framework for conditional indep en

B D iAD

thus corresp onding to the standard probabilistic def

dence The main dierence is that their combination

inition of conditional indep endence for this case Let

is dened also for noncompatible valuations How

us call these semigraphoids as triplets of pairwise dis

ever Shenoy also derived graphoid prop erties

joint subsets of N discrete probabilistic models

from his axioms for valuations

GENERAL CASE

PROBABILISTIC FRAMEWORK

Now for every i N a measurable space X X is

i i

given and an ob ject with domain D N is a prob

The classic example of ob jects satisfying our axioms is

ability measure P on the pro duct space X X

D D

Q

given by probability measures on Cartesian pro ducts

X X The pro jection of P onto E N is the

i i

iD

of arbitrary measurable spaces We start with the im

marginal measure of P on X X

E D E D

p ortant sp ecial case of discrete probability measures

E

then treat the general case Throughout we take N

P T P T X for every T X

E D

D nE

to b e the class of nite subsets of N

E

Of course P P if D n E An example showing

that weak compatibility may not imply strong com

patibility in this general framework that is C C

DISCRETE CASE

is given in App endix B Example B

For each i N we are given a nonempty nite set

Q

The denition of conditional pro duct is now more tech

X X For nonempty D N we dene X

i i D

iD

nical for relevant background see for example Neveu

while X is taken to b e some xed set fg

Let P having domain E N and Q hav

An ob ject with domain D is a distribution over D ie

ing domain F N b e weakly compatible Given

P

a nonnegative function p on X such that f px

D

T X where E n F by a representative of the

E nF

x X g

D

of T given X induced by

E F

P is understo o d any X measurable nonnegative

E F

The pro jection of p over D onto E N is the marginal

E

function P Tj on X such that

E F

distribution p on X dened by the formula

E D

Z

E F

X Y

P Tju dP u P T U

E

X g p y f px y x

i

uU

iD nE

for every U X Observe that in the discrete case

E F

one has

pt u

for every y X with obvious mo dication if E

E D

P ftgju

E

E F

D Of course p p if D n E

p u

E F

The conditional pro duct of a distribution p over E N for t X and u X with p u For

E F

E nF

and a distribution q over F N will b e dened for each T such a representative exists and any two rep

every pair of weakly compatible distributions so that resentatives can dier only on a set in X of P

E F

C D in this case Supp osing p is dened on X and probability However it is not always p ossible to E

much simpler The corresp onding concept of condi choose representatives for all T X to ensure

E nF

tional indep endence will turn out to b e variation inde additivity over T

pendence Dawid which arises naturally in the

Conditional probability QV j of V X given

F nE

context of relational databases and has applications

X induced by Q is dened analogously Then one

E F

in the statistical analysis of graphical mo dels Dawid

can introduce the following set function

and Lauritzen

Z

E F

P Tju QV ju dP u R T U V

We again have a space X for every i N but the

i

uU

algebra X is no longer required An ob ject S with

i

for T X U X V X Of course

E F

E nF F nE

domain D N will now b e an arbitrary subset of

Cartesian pro ducts over the empty set are omitted and

X We start by dening pro jection and conditional

D

E F E F

P Q is the common pro jection of P and Q

pro duct for p oints Thus let x x i D Then for

i

E

onto E F

E N we dene its pro jection onto E to b e x x

i

i D E We say that two p oints x x i E

i

Equation denes R only on the sub class S of sets

and y y i F are compatible if x y for every

i i i

in X of the form T U V as describ ed ab ove

E F

i E F ie they have the same pro jections onto

Then R is nitely additive but need not b e additive

E F and in this case dene their conditional pro duct

on S If it is additive it can b e uniquely extended

x y as z z i E F where z x if i E z

i i i i

to a additive on X which

E F

y if i F Then pro jection of an ob ject S is dened

i

we also denote by R In this case P Q D and we

E E

p ointwise S fx x S g If S and T are two

dene the conditional pro duct P Q R Otherwise

ob jects with resp ective domains E and F they will

P Q D and P Q is not dened

b e compatible if their pro jections onto E F coincide

An example showing that strong compatibility do es

and in this case we take S T D dening S T

not imply the existence of the conditional pro duct ie

fx y x S y T x and y are compatibleg It is

D C is given in App endix B Example B

then not dicult to verify

With the ab ove denition the corresp onding concept

Prop osition The axioms of x hold for the

of conditional indep endence is given by for pairwise

above dened conditional product of sets

disjoint A B C N AB j C P if

for T X V X U X

A B C

The corresp onding denition of conditional inde

R

C

p endence is given by for disjoint A B C N

P Tju P V ju dP u P T U V

uU

C A B

AB j C S if for each z S fx x x

We note by Observation that AB j C P Q

C

S x z g is a Cartesian pro duct That is to say as

for P Q D dP A C dQ B C since

a p oint varies in S sub ject to having a given pro jec

AC B C

P Q P P Q Q

tion onto C there are no constraints relating its pro

jections onto A and onto B By Prop osition this

We assert without pro of that equivalent statements

concept of variation indep endence denes a disjoint

to are

semigraphoid on N

C

P T V j u P T j uP V j u as P

B C

P T j u v P T j u as P

GRAPHICAL FRAMEWORK

AC

P V j t u P V j u as P

Further and are equivalent to the existence

In this Section we explore two sp ecic graphical frame

of a X measurable representative of P T j u v or of

C

works which are widely used in Articial Intelligence

P V j t u resp ectively

undirected graphs and directed acyclic graphs

The following prop osition is proved in App endix A

UNDIRECTED GRAPHS

Prop osition The axioms of x hold for the

above dened conditional product of probability mea

An undirected graph G over a nite set of no des D N

sures

is sp ecied by a collection L of twoelement subsets of

D which are called edges We call D the no deset and

It now follows from Prop ositions and that gen

L the edgeset of G and write G D L A path in G

eral probabilistic conditional indep endence as dened

is a sequence of distinct no des w w n D

1 n

by ab ove via conditional pro ducts induces a dis

such that fw w g L for i n We say

i i+1

joint semigraphoid which we may term a probabilistic

that a triplet hA B jC i of pairwise disjoint nite sub

model

sets of N is represented on G and write AB j C G

if for every path w w in G with w A and

1 n 1

SETTHEORETIC FRAMEWORK

w B there exists i n with w C It is

n i

no problem to verify that j G forms a disjoint

semigraphoid over N see for example Pearl In this Section we consider a framework which is in

Let us call such semigraphoids UGmodels over N some ways analogous to the probabilistic one but

product operations on undirected graphs

a a

f f

G G

2 1

The reader may incline to conclude that conditional

pro duct op eration cannot induce UGmo dels at all

f f f f

b d b d

However this is not true The reason is that the

framework of undirected graphs can b e considered as a

subframework of the discrete probabilistic framework

f f

Geiger and Pearl showed that for every UG

c c

mo del over D N there exists a discrete probability

distribution over D inducing it see also Studen y and

Figure Two Undirected Graphs

Bouckaert Prop osition says that every

such discrete probabilistic mo del can b e dened by

means of a conditional pro duct on discrete probabil

a a

f f

ity distributions

An Alternative Construction

f f

b d

Although we have seen that there is no way of den

ing op erations of pro jection and conditional pro duct

f f

on undirected graphs so as to reconstruct the stan

c c

dard denition of graphical conditional indep endence

separation other constructions are p ossible lead

Figure Their Common Pro jections

ing to new concepts of graphical separation Thus let

G D L b e an undirected graph on D N For

A D the pro jection of G onto A is just the induced

A

A natural question arises Is it p ossible to dene the

subgraph G A L A A while for more gen

A AD

ab ove mentioned conditional indep endence predicate

eral A we dene G G Then two graphs G and

by means of a suitable conditional pro duct op era

H are compatible if they have exactly the same edges

tion on undirected graphs The answer is negative

b etween all no des common to b oth their domains In

this case we take G H D and dene their condi

First supp ose for contradiction that separation in

tional pro duct G H as the graph whose no deset and

undirected graphs can b e equivalently dened by

edgeset are obtained as the unions of the corresp ond

means of some pro jection and conditional pro duct sat

ing no desets and edgesets of G and H It is readily

E

isfying the axioms from x Let G b e the pro jection

seen that all the axioms of Section are satised for

of an undirected graph G over D N onto E N

these denitions Corresp ondingly we have the fol

E

Then Observation and P imply AB j C G

lowing denition of conditional indep endence with re

i AB j C G for every triplet hA B jC i of pair

sp ect to a graph G D L for disjoint A B C N

wise disjoint nite sets of A B C E D Observe

AB j C G if there is no edge in L containing one el

that fu v g is an edge in an undirected graph H over

ement in A and the other in B By Prop osition

E D i fugfv g j E D n fu v g H Hence

this denition yields a disjoint semigraphoid over N

E

fu v g E D is an edge in G i there exists a path

However it is distinct from any of the usual forms mo

u w w v n in G with w D n E

1 n i

tivated by probabilistic indep endence In particular

E

for i n Thus the pro jection G is uniquely

we see that for given A and B AB j C holds or fails

determined Note that pro jection op eration dened in

simultaneously for every conditioning set C so long

this way satises PP

only as the disjointness requirement is maintained

To show that no reasonable conditional pro duct op er

DIRECTED ACYCLIC GRAPHS

ation can b e introduced within this framework con

sider the two graphs over fa b c dg from Figure

A directed graph H over a nite set of no des D N

Evidently fbgfdg j fa cg G for i In case

i

is sp ecied by a collection A of ordered pairs u v

can b e dened by means of a conditional pro d

of distinct no des called arrows A descending path

uct op eration deduce from the denition in x

in H is a sequence of distinct no des w w n

facdg fabcg

1 n

for i However the G G G

i

i i

such that w w A for i n It

i i+1

corresp onding pro jections of G onto fa b cg and onto

i

is called a directed cycle if moreover w w A

n 1

fa c dg coincide they are as in Figure Hence a

A directed acyclic graph is a directed graph without

contradictory conclusion G G is derived Thus

1 2

directed cycles

we have

Testing whether a triplet hA B jC i of pairwise disjoint

nite subsets of N is represented in such a graph is Consequence

more complicated than in the undirected case Let us Conditional independence for undirected graphs can

describ e the moralization criterion of Lauritzen et al not be dened by means of projection and conditional

fb dg is an edge in K The fact fbgfeg j H im

a c e

v f v

plies that fb eg is not an edge in K Finally the fact

fbgfeg j L H for fdg L fa dg implies that

H

b d and e d are arrows in K However by the same

R R

v v

consideration interchanging a with e and b with d de

b d

rive that d b and a b are arrows in K Then the

conclusion that b oth b d and d b are arrows in K

contradicts the assumption that K is acyclic

Figure A Directed Acyclic Graph

It follows from Lemma and Observation that

which is equivalent to the dseparation criterion

a suitable pro jection of H onto F cannot b e dened

of Pearl First consider the set E of ancestors

Thus we have

of A B C that is the set of no des u D such

that there exists a descending path u w w

Consequence Conditional independence for di

1 n

A B C n in H An undirected graph G

rected acyclic graphs cannot be dened by means of

over E called the moral graph of H is constructed

projection and conditional product operations on di

as follows fu v g L i u v A or v u A or

rected acyclic graphs

there exists w E with u w A and v w A

The triplet hA B jC i is represented in H denoted by

Again as in the undirected case the framework of

AB j C H if it is represented in the moral graph

directed acyclic graphs can b e considered as a sub

G in the sense describ ed in x Again the reader

framework of the discrete probabilistic framework

can verify that j H determines a disjoint semi

The result that every DAGmodel is a discrete proba

graphoid over N Let us call such semigraphoids

bilistic mo del is given by Geiger and Pearl see

DAGmodels

also Studen y and Bouckaert

One can raise a question analogous to that in the previ

ous case Is there any conditional pro duct op eration on

CONCLUSION

directed acyclic graphs inducing the conditional inde

p endence predicate ab ove The answer is again nega

We have introduced an abstract concept of conditional

tive The reason is even more basic than in undirected

pro duct and shown how it can b e used to derive

case no reasonable pro jection op eration exists To see

an induced concept of conditional indep endence

this consider the graph H of Figure and ask what

Conversely if we start with a denition of ob ey

could b e the pro jection of H onto F fa b d eg By

ing the semigraphoid axioms we can search for an

Observation the DAGmodel over F induced by

asso ciated conditional pro duct in particular for

this pro jection is uniquely determined However we

D with d A d B would

A B

observe the following

have to satisfy and AB j C see

Observation As we have seen in the framework of

Lemma There is no directed acyclic graph K

graphical mo dels this is not always p ossible In other

over fa b d eg such that for every triplet A B C

contexts the required construction may b e p ossible

fa b d eg of pairwise disjoint sets AB j C K i

but nonunique

AB j C H

We have developed the theory of semigraphoids and

conditional pro ducts for the sp ecial case of domains Pro of The argument is based on two observations

which are subsets of some given index set N Cor concerning an arbitrary directed acyclic graph K over

resp ondingly our denitions in the probabilistic and F The rst observation is that fu v g is an edge in

settheoretic frameworks have b een based on sample K that is either u v or v u is an arrow in K i

spaces which are Cartesian pro ducts In fact it is p ossi fugfv g j L K for every L F n fu v g Indeed

ble to develop the theory in a still more general frame to show suciency of this condition take as L the set

work in which we need only require that the class of ancestors of fu v g in K excluding u and v The

of p ossible domains b e an abstract join semilattice second observation is that whenever fu w g and fv w g

Birkho are edges in K but fu v g is not an edge in K then

u w and v w are simultaneously arrows in K i

The general abstract algebraic structures underlying

fugfv g j L K for every fw g L F n fu v g

conditional indep endence would seem to have consider

Indeed to show suciency of the condition take as L

able indep endent mathematical interest and promise

the set of ancestors of fu v w g in K excluding u and

to richly repay further study Also they bring a uni

v

fying p oint of view to the study of many application

areas and it will b e fruitful to identify new applica Supp ose for contradiction the existence of a graph

tions b oth within and b eyond the motivating areas K over fa b d eg satisfying the required conditions

of probability and other uncertainty formalisms and Then the fact fdgfeg j L H for every L fa bg

graph theory In particular in further work we plan implies that fd eg is an edge in K Similarly the fact

to investigate the existence nature and prop erties of fbgfdg j L H for every L fa eg implies that

measurable representative of R T j u z and thus a X conditional pro ducts for a number of sp ecial problem

C

measurable representative of S T j u z Hence there areas of natural interest including

exists a X measurable representative of S T j u w z

C

Possibility theory and Sp ohns theory of

which is equivalent to AB D j C S Since

AC B C D

functions as mentioned in Shenoy

S P S Q this implies S P Q

Meta Markov models and hyper Markov models

Dawid and Lauritzen These are formed

B TWO EXAMPLES Studen y

by suitably combining the concepts of probabilis

tic x and settheoretic x conditional pro d

To give examples that C C and C D in the

uct The combination pro cess involved can b e

framework describ ed in x we utilize the following

abstracted to apply to conditional pro ducts more

lemma

generally

Lemma B Let T A b e a measurable space and

DempsterShafer belief functions Shafer

B A a sub algebra such that the diagonal ft t

Again suitable denitions of conditional pro d

t T g is A B measurable Let us put X X

uct and conditional indep endence here will in

1 1

X X T A X X T B

volve combining probabilistic and settheoretic

3 3 2 2

concepts

i Every probability measure P on X X

1 2

Semigraphoids induced by imsets sup ermo du

X X X X whose marginals on X X

3 1 2 3 1 2

lar integervalued set functions on subsets of N

and X X are concentrated on their diagonals

2 3

Studen y

is concentrated on the diagonal D ft t t

t T g

Completely general semigraphoids over N Is

there any reasonable denition of conditional

ii Every probability measure on T A induces

pro duct for semigraphoids Or p erhaps for in

by the formula

teresting sub classes such as those arising from

probabilistic mo dels

P A B C A B C

for A X B X A X a probability measure

1 2 3

A PROOF OF PROPOSITION

P on X X X X X X concentrated

1 2 3 1 2 3

on the diagonal

T and T are immediate from the denition For

F

iii Conversely every such measure can b e introduced

T take P to have domain E and Q P where

in this way

without loss of generality F E In V X

f fgg and we have Q j u as Q Qfg j u

iv The marginal of P on X and on X is the

1 3

as Q We can regard R as dened on a sub class

marginal of P on X is the restriction of to B

2

of X on identifying T U fg with T U and

E

R

F

P TjudP u with U X then R T U

F

uU

Pro of For i realize that b oth D ft t v

1

F F

P T U So clearly P P D and and P P

t v T g and D fv t t t v T g b elong

2

P ie T holds

to X X X Of course D D D and

1 2 3 1 2

the assumption P D P D imply P D

1 2

Now supp ose P Q D with dP A C

Consider the mapping t t t t from T into

dQ B C D where A B C D are pairwise dis

AC B C D

X X X The reader can easily verify that it is

1 2 3

joint Let S P Q so that S P S

a measurable onetoone transformation of T A into

Q Then AB D j C S Using

B C D

X X X X X X realize that the inverse

1 2 3 1 2 3

we thus have S T j u w z S T j u as S

image of A B C is A B C whose inverse transfor

with u X w z X X X

C B D B D

mation is measurable as well the image of A can b e

Equivalently S T j u w z is more precisely has a

written as A T T D Since the image of T is

representative that is X measurable It readily

C

D probability measures on T A are transformed to

follows that S T j u z is X measurable so that

C

AC D AC D C D

probability measures on X X X X X X

1 2 3 1 2 3

AD j C S ie P Q P Q

concentrated on D Thus b oth ii and iii are evi

viz T holds and that S T j u w z is X

C D

dent iv follows from the formula in ii

measurable so AB j C D S ie P Q

AC D

P Q Q viz T holds

Let us put T h i and consider the algebra B of

For verication of T take P Q with dP A

Borel subsets of T Let us denote by the Leb esgue

C dQ B C D where A B C D are pair

measure on T B It was shown in Halmos

C D

wise disjoint Let R P Q S R Q

Theorem E in x that there exists a set M T

AC D

so that S R Then AB j C D S

such that M T n M Here denotes

and using we have that S T j u w z

Leb esgue inner measure dened by the formula

S T j u z as S with u X w X z X

C B D

However AD j C R implies that there exists a X M supf K K M K B g

C

gives a representative of the conditional probability Put A f E M F n M E F B g it is the

of G given X B induced by P verify using algebra generated by B fMg Observe that whenever

2

1

and with A similar conclusion holds

E M F n M E M F n M for E E F F B then

2

1 1

for Q Put T h M h i n M X

1

EE T n M and FF M and therefore EE

2 2

1 1

i M h n M X U h i X V h

3 2

FF In particular for every h i the set

2 2

1 1

function on A dened by

Then pTj and q V j by the formula ab ove

2 2

1

and implies R T U V T T U V

G E F

4

However since T V gives a contradictory

where G E M F n M E F B is well

conclusion R T U V Therefore P and Q have

dened The reader can show by the pro cedure in

no conditional pro duct

the pro of of Theorem A x of Halmos that

is a probability measure on T A Evidently

Acknowledgments

the restriction of to B is for every h i

is concentrated on M and is concentrated on

1 0

Milan Studen ys work was supp orted by grants GACR

T n M Put X X T A X X T B

1 1 2 2

n MSMT n and GAAVCR n

X X T A and introduce the measure R on

3 3

A

X X X X X X for h i by the

1 2 3 1 2 3

formula

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