Definition. The pair {X,Y}{X,Y} is conditionally independent, givenZ, denoted
{X,Y} ci |Z{X,Y} ci |Z iff
E
[
I
M
(
X
)
I
N
(
Y
)
|
Z
]
=
E
[
I
M
(
X
)
|
Z
]
E
[
I
N
(
Y
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
,
N
E
[
I
M
(
X
)
I
N
(
Y
)
|
Z
]
=
E
[
I
M
(
X
)
|
Z
]
E
[
I
N
(
Y
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
,
N
(1)An arbitrary class {Xt:t∈T}{Xt:t∈T} of random vectors is conditionally independent, give Z,
iff such a product rule holds for each finite subclass or two or more members of the class.
Remark. The expression “for all Borel sets M,NM,N,” here and elsewhere, implies the sets are
on the appropriate codomains. Also, the expressions below “for all Borel functions g,” etc.,
imply that the functions are real-valued, such that the indicated expectations are finite.
The following are equivalent. Each is necessary and sufficient that {X,Y} ci |Z{X,Y} ci |Z.
- (CI1) :
E
[
I
M
(
X
)
I
N
(
Y
)
|
Z
]
=
E
[
I
M
(
X
)
|
Z
]
E
[
I
N
(
Y
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
,
N
E
[
I
M
(
X
)
I
N
(
Y
)
|
Z
]
=
E
[
I
M
(
X
)
|
Z
]
E
[
I
N
(
Y
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
,
N
- (CI2) :
E
[
I
M
(
X
)
|
Z
,
Y
]
=
E
[
I
M
(
X
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
E
[
I
M
(
X
)
|
Z
,
Y
]
=
E
[
I
M
(
X
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
- (CI3) :
E
[
I
M
(
X
)
I
Q
(
Z
)
|
Z
,
Y
]
=
E
[
I
M
(
X
)
I
Q
(
Z
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
,
Q
E
[
I
M
(
X
)
I
Q
(
Z
)
|
Z
,
Y
]
=
E
[
I
M
(
X
)
I
Q
(
Z
)
|
Z
]
a
.
s
.
for
all
Borel
sets
M
,
Q
- (CI4) : E[IM(X)IQ(Z)|Y]=E{E[IM(X)IQ(Z)|Z]|Y}a.s.forallBorelsetsM,QE[IM(X)IQ(Z)|Y]=E{E[IM(X)IQ(Z)|Z]|Y}a.s.forallBorelsetsM,Q
****
- (CI5) :
E
[
g
(
X
,
Z
)
h
(
Y
,
Z
)
|
Z
]
=
E
[
g
(
X
,
Z
)
|
Z
]
E
[
h
(
Y
,
Z
)
|
Z
]
a
.
s
.
for
all
Borel
functions
g
,
h
E
[
g
(
X
,
Z
)
h
(
Y
,
Z
)
|
Z
]
=
E
[
g
(
X
,
Z
)
|
Z
]
E
[
h
(
Y
,
Z
)
|
Z
]
a
.
s
.
for
all
Borel
functions
g
,
h
- (CI6) :
E
[
g
(
X
,
Z
)
|
Z
,
Y
]
=
E
[
g
(
X
,
Z
)
|
Z
]
a
.
s
.
for
all
Borel
functions
g
E
[
g
(
X
,
Z
)
|
Z
,
Y
]
=
E
[
g
(
X
,
Z
)
|
Z
]
a
.
s
.
for
all
Borel
functions
g
- (CI7) : For any Borel function g, there exists a Borel function eg such that
E[g(X,Z)|Z,Y]=eg(Z)a.s.E[g(X,Z)|Z,Y]=eg(Z)a.s.
(2) - (CI8) : E[g(X,Z)|Y]=E{E[g(X,Z)|Z]|Y}a.s.forallBorelfunctionsgE[g(X,Z)|Y]=E{E[g(X,Z)|Z]|Y}a.s.forallBorelfunctionsg
****
- (CI9) : {U,V} ci |Z{U,V} ci |Z, where U=g(X,Z)U=g(X,Z) and V=h(Y,Z)V=h(Y,Z), for any Borel
functions g,hg,h.
Additional properties of conditional independence
- (CI10): {X,Y} ci |Z{X,Y} ci |Z implies {X,Y} ci |(Z,U){X,Y} ci |(Z,U), {X,Y} ci |(Z,V){X,Y} ci |(Z,V), and {X,Y} ci |(Z,U,V){X,Y} ci |(Z,U,V),
where U=h(X)U=h(X) and V=k(Y)V=k(Y), with h,kh,k Borel.
- (CI11): {X,Z} ci |Y{X,Z} ci |Y and {X,W} ci |(Y,Z){X,W} ci |(Y,Z) iff {X,(Z,W)} ci |Y{X,(Z,W)} ci |Y.
- (CI12): {X,Z} ci |Y{X,Z} ci |Y and {(X,Y),W} ci |Z{(X,Y),W} ci |Z implies
{X,(Z,W)} ci |Y{X,(Z,W)} ci |Y.
- (CI13): {X,Y}{X,Y} is independent and {X,Z} ci |Y{X,Z} ci |Y iff
{X,(Y,Z)}{X,(Y,Z)} is independent.
- (CI14): {X,Y} ci |Z{X,Y} ci |Z implies E[g(X,Y)|Y=u,Z=v]=E[g(X,u)|Z=v]a.s.[PYZ]E[g(X,Y)|Y=u,Z=v]=E[g(X,u)|Z=v]a.s.[PYZ]
- (CI15): {X,Y} ci |Z{X,Y} ci |Z implies
-
E[g(X,Z)h(Y,Z)]=E{E[g(X,Z)|Z]E[h(Y,Z)|Z]}=E[e1(Z)e2(Z)]E[g(X,Z)h(Y,Z)]=E{E[g(X,Z)|Z]E[h(Y,Z)|Z]}=E[e1(Z)e2(Z)]
-
E[g(Y)|X∈M]P(X∈M)=E{E[IM(X)|Z]E[g(Y)|Z]}E[g(Y)|X∈M]P(X∈M)=E{E[IM(X)|Z]E[g(Y)|Z]}
- (CI16): {(X,Y),Z} ci |W{(X,Y),Z} ci |W iff E[IM(X)IN(Y)IQ(Z)|W]=E[IM(X)IN(Y)|W]E[IQ(Z)|W]a.s.E[IM(X)IN(Y)IQ(Z)|W]=E[IM(X)IN(Y)|W]E[IQ(Z)|W]a.s.
for all Borel sets M,N,QM,N,Q
