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TH
E M
AT
HE
MA
TIC
S
OF
CA
US
AL
IN
FE
RE
NC
E
IN S
TA
TIS
TIC
S
Ju
de
a P
ea
rl
De
pa
rtm
en
t o
f C
om
pu
ter
Sc
ien
ce
UC
LA
•S
tati
sti
ca
l v
s.
Ca
us
al
Mo
de
ling
:
dis
tin
cti
on
an
d m
en
tal
ba
rrie
rs
•N-R
vs
. s
tru
ctu
ral
mo
de
l:
str
en
gth
s a
nd
we
ak
ne
ss
es
•F
orm
al
se
ma
nti
cs
fo
r c
ou
nte
rfa
ctu
als
:
de
fin
itio
n,
ax
iom
s,
gra
ph
ica
l re
pre
se
nta
tio
ns
•G
rap
hs
an
d A
lge
bra
: S
ym
bio
sis
tra
ns
lati
on
an
d a
cc
om
plis
hm
en
ts
OU
TL
INE
TR
AD
ITIO
NA
L S
TA
TIS
TIC
AL
INF
ER
EN
CE
PA
RA
DIG
M
Da
ta
Infe
ren
ce
Q(P
)
(As
pe
cts
of P
)
P
Jo
int
Dis
trib
uti
on
e.g
.,
Infe
r w
he
the
r c
us
tom
ers
wh
o b
ou
gh
t p
rod
uc
t A
wo
uld
als
o b
uy
pro
du
ct B
.
Q=
P(B | A
)
Wh
at
ha
pp
en
s w
he
n P
ch
an
ge
s?
e.g
.,
Infe
r w
he
the
r c
us
tom
ers
wh
o b
ou
gh
t p
rod
uc
t A
wo
uld
sti
ll b
uy
Aif
we
we
re t
o d
ou
ble
th
e p
ric
e.
FR
OM
ST
AT
IST
ICA
L T
O C
AU
SA
L A
NA
LY
SIS
:
1.
TH
E D
IFF
ER
EN
CE
S
Pro
ba
bili
ty a
nd
sta
tis
tic
s d
ea
l w
ith
sta
tic
re
lati
on
s
Da
ta
Infe
ren
ce
Q(P
′)(A
sp
ec
ts o
f P
′)
P′
Jo
int
Dis
trib
uti
on
P
Jo
int
Dis
trib
uti
on c
ha
ng
e
FR
OM
ST
AT
IST
ICA
L T
O C
AU
SA
L A
NA
LY
SIS
:
1.
TH
E D
IFF
ER
EN
CE
S
Note
: P
′(v)
≠P (v
| price = 2
)
Pd
oe
s n
ot
tell
us
ho
w i
t o
ug
ht
to c
ha
ng
e
e.g
. C
uri
ng
sy
mp
tom
s v
s.
cu
rin
g d
ise
as
es
e.g
. A
na
log
y:
me
ch
an
ica
l d
efo
rma
tio
n
Wh
at
rem
ain
s i
nv
ari
an
t w
he
n P
ch
an
ge
s s
ay
, to
sa
tis
fy
P′(price
=2)=
1
Da
ta
Infe
ren
ce
Q( P
′)(A
sp
ec
ts o
f P
′)
P′
Jo
int
Dis
trib
uti
on
P
Jo
int
Dis
trib
uti
on c
ha
ng
e
FR
OM
ST
AT
IST
ICA
L T
O C
AU
SA
L A
NA
LY
SIS
:
1.
TH
E D
IFF
ER
EN
CE
S (
CO
NT
)
CA
US
AL
Sp
uri
ou
s c
orr
ela
tio
n
Ra
nd
om
iza
tio
n
Co
nfo
un
din
g /
Eff
ec
t
Ins
tru
me
nt
Ho
ldin
g c
on
sta
nt
Ex
pla
na
tory
va
ria
ble
s
ST
AT
IST
ICA
L
Re
gre
ss
ion
As
so
cia
tio
n /
In
de
pe
nd
en
ce
“Co
ntr
olli
ng
fo
r” /
Co
nd
itio
nin
g
Od
d a
nd
ris
k r
ati
os
Co
llap
sib
ility
1.
Ca
us
al
an
d s
tati
sti
ca
l c
on
ce
pts
do
no
t m
ix.
2.
3.
4.
CA
US
AL
Sp
uri
ou
s c
orr
ela
tio
n
Ra
nd
om
iza
tio
n
Co
nfo
un
din
g /
Eff
ec
t
Ins
tru
me
nt
Ho
ldin
g c
on
sta
nt
Ex
pla
na
tory
va
ria
ble
s
ST
AT
IST
ICA
L
Re
gre
ss
ion
As
so
cia
tio
n /
In
de
pe
nd
en
ce
“Co
ntr
olli
ng
fo
r” /
Co
nd
itio
nin
g
Od
d a
nd
ris
k r
ati
os
Co
llap
sib
ility
1.
Ca
us
al
an
d s
tati
sti
ca
l c
on
ce
pts
do
no
t m
ix.
4.
3.
Ca
us
al
as
su
mp
tio
ns
ca
nn
ot
be
ex
pre
ss
ed
in
th
e m
ath
em
ati
ca
l
lan
gu
ag
e o
f s
tan
da
rd s
tati
sti
cs
.
FR
OM
ST
AT
IST
ICA
L T
O C
AU
SA
L A
NA
LY
SIS
:
1.
TH
E D
IFF
ER
EN
CE
S (
CO
NT
)
2.
No
ca
us
es
in
–n
o c
au
se
s o
ut
(Ca
rtw
rig
ht,
19
89
)
sta
tis
tic
al
as
su
mp
tio
ns
+ d
ata
ca
us
al
as
su
mp
tio
ns
ca
us
al
co
nc
lus
ion
s⇒}
4.
No
n-s
tan
da
rd m
ath
em
ati
cs
:
a)
Str
uc
tura
l e
qu
ati
on
mo
de
ls (
Wri
gh
t, 1
92
0;
Sim
on
, 1
96
0)
b)
Co
un
terf
ac
tua
ls (
Ne
ym
an
-Ru
bin
(Yx),
Le
wis
(x
Y))
CA
US
AL
Sp
uri
ou
s c
orr
ela
tio
n
Ra
nd
om
iza
tio
n
Co
nfo
un
din
g /
Eff
ec
t
Ins
tru
me
nt
Ho
ldin
g c
on
sta
nt
Ex
pla
na
tory
va
ria
ble
s
ST
AT
IST
ICA
L
Re
gre
ss
ion
As
so
cia
tio
n /
In
de
pe
nd
en
ce
“Co
ntr
olli
ng
fo
r” /
Co
nd
itio
nin
g
Od
d a
nd
ris
k r
ati
os
Co
llap
sib
ility
1.
Ca
us
al
an
d s
tati
sti
ca
l c
on
ce
pts
do
no
t m
ix.
3.
Ca
us
al
as
su
mp
tio
ns
ca
nn
ot
be
ex
pre
ss
ed
in
th
e m
ath
em
ati
ca
l
lan
gu
ag
e o
f s
tan
da
rd s
tati
sti
cs
.
FR
OM
ST
AT
IST
ICA
L T
O C
AU
SA
L A
NA
LY
SIS
:
1.
TH
E D
IFF
ER
EN
CE
S (
CO
NT
)
2.
No
ca
us
es
in
–n
o c
au
se
s o
ut
(Ca
rtw
rig
ht,
19
89
)
sta
tis
tic
al
as
su
mp
tio
ns
+ d
ata
ca
us
al
as
su
mp
tio
ns
ca
us
al
co
nc
lus
ion
s⇒}
TW
O P
AR
AD
IGM
S F
OR
CA
US
AL
IN
FE
RE
NC
E
Ob
se
rve
d:
P(X, Y, Z,...)
Co
nc
lus
ion
s n
ee
de
d:
P(Y
x=y)
, P
(Xy=x
| Z=z)
...
Ho
w d
o w
e c
on
ne
ct
ob
se
rva
ble
s, X,Y,Z
,…
to c
ou
nte
rfa
ctu
als
Yx, X
z, Z
y,…
?
N-R m
od
el
Co
un
terf
ac
tua
ls a
re
pri
mit
ive
s,
ne
w v
ari
ab
les
Su
pe
r-d
istr
ibu
tio
n
P*(X, Y,…
, Yx, X
z,…
)
X, Y, Z
co
ns
tra
in
Yx, Z
y,…
Str
uc
tura
l m
od
el
Co
un
terf
ac
tua
ls a
re
de
riv
ed
qu
an
titi
es
Su
bs
cri
pts
mo
dif
y t
he
dis
trib
uti
on
P(Y
x=y)
= P
x(Y=y)
“SU
PE
R”
DIS
TR
IBU
TIO
N
IN N-R
MO
DE
L
X 0 0 0 1
Y 0 1 0 0
Yx=
0
0 1 1 1
Z 0 1 0 0
Yx=
1
1 0 0 0
Xz=
0
0 1 0 0
Xz=
1
0 0 1 1
Xy=
0⋅⋅⋅
0⋅⋅⋅
1⋅⋅⋅
1⋅⋅⋅
0⋅⋅⋅
U u1
u2
u3
u4
inc
on
sis
ten
cy
:
x
= 0
⇒Yx=
0=
YY = xY
1+ (
1-x
) Y
0
yx
zx
xyxz
yx
ZX
Y
XZ
yY
P
ZY
ZY
ZY
XP
|
),
|(
*
...)
,...
,...
,,...
,,
(*
⊥⊥
=
:D
efi
ne
s
Is c
on
sis
ten
cy
th
e o
nly
co
nn
ec
tio
n b
etw
ee
n⊥
Yx
⊥X
| Z
jud
gm
en
tal
&o
pa
qu
e
Try
it:
X →
Y →
Z?
op
aq
ue
⊥Yx
⊥X
| Z
jud
gm
en
tal
&
TY
PIC
AL
IN
FE
RE
NC
E
IN N-R
MO
DE
LF
ind
P*(Y
x=y)
giv
en
co
va
ria
te Z
, ∑∑∑∑
=
==
==
==
=
zzzx
zx
x
zP
zx
yP
zP
zx
yY
P
zP
zx
yY
P
zP
zy
YP
yY
P
)(
),
|(
)(
),
|(
*
)(
),
|(
*
)(
)|
(*
)(
*
Pro
ble
ms
:
1)
X, Y
an
d Y
x?
2)
As
su
me
co
ns
iste
nc
y:
X=x ⇒
Yx=Y
As
su
me
ig
no
rab
ility
:
Yx
⊥X | Z
⊥
Is c
on
sis
ten
cy
th
e o
nly
co
nn
ec
tio
n b
etw
ee
n
Da
ta
Infe
ren
ce
Q(M
)
(As
pe
cts
of M
)
Da
ta
Ge
ne
rati
ng
Mo
de
l
M–
Ora
cle
fo
r c
om
pu
tin
g a
ns
we
rs t
o Q
’s.
e.g
.,
Infe
r w
he
the
r c
us
tom
ers
wh
o b
ou
gh
t p
rod
uc
t A
wo
uld
sti
ll b
uy
Aif
we
we
re t
o d
ou
ble
the
pri
ce
.
Jo
int
Dis
trib
uti
on
TH
E S
TR
UC
TU
RA
L M
OD
EL
PA
RA
DIG
M
Z
YX
INP
UT
OU
TP
UT
FA
MIL
IAR
CA
US
AL
MO
DE
L
OR
AC
LE
FO
R M
AN
IPIL
AT
ION
ST
RU
CT
UR
AL
CA
US
AL
MO
DE
LS
De
fin
itio
n:
A s
tru
ctu
ral
ca
us
al
mo
de
lis
a 4
-tu
ple
⟨ ⟨⟨⟨V,U, F, P
(u)⟩ ⟩⟩⟩
, w
he
re
•V
= {V
1,...,V
n}
are
ob
se
rva
ble
va
ria
ble
s
•U
={U
1,...,U
m}
are
ba
ck
gro
un
d v
ari
ab
les
•F
= {f 1,...,f n
}a
re f
un
cti
on
s d
ete
rmin
ing
V,
v i=
fi(v,
u)
•P
(u)
is a
dis
trib
uti
on
ov
er U
P(u
)a
nd
Fin
du
ce
a d
istr
ibu
tio
n P
(v)
ov
er
ob
se
rva
ble
va
ria
ble
s
ST
RU
CT
UR
AL
MO
DE
LS
AN
D
CA
US
AL
DIA
GR
AM
S
Th
e a
rgu
me
nts
of
the
fu
nc
tio
nsv i
= fi(v,u)
de
fin
e a
gra
ph
v i=
fi(pai,u
i)PAi⊆V
\Vi
Ui⊆U
Ex
am
ple
: P
ric
e –
Qu
an
tity
eq
ua
tio
ns
in
ec
on
om
ics
U1
U2
IW
QP
PAQ
22
2
11
1
uw
dq
bp
ui
dp
bq
++
=
++
=
U1
U2
IW
QP
22
2
11
1
uw
dq
bp
ui
dp
bq
++
=
++
=
Le
t X
be
a s
et
of
va
ria
ble
s i
n V
.
Th
e a
cti
on
do(x
)s
ets
Xto
co
ns
tan
ts x
reg
ard
les
s o
f
the
fa
cto
rs w
hic
h p
rev
iou
sly
de
term
ine
d X
.
do
(x)
rep
lac
es
all
fun
cti
on
s fi
de
term
inin
g X
wit
h t
he
co
ns
tan
t fu
nc
tio
nsX=x,
to
cre
ate
a m
uti
late
d m
od
elM
x
ST
RU
CT
UR
AL
MO
DE
LS
AN
D
INT
ER
VE
NT
ION
U1
U2
IW
QP
P = p
0
0
22
2
11
1 pp
uw
dq
bp
ui
dp
bq
=
++
=
++
=
Mp
Le
t X
be
a s
et
of
va
ria
ble
s i
n V
.
Th
e a
cti
on
do(x
)s
ets
Xto
co
ns
tan
ts x
reg
ard
les
s o
f
the
fa
cto
rs w
hic
h p
rev
iou
sly
de
term
ine
d X
.
do
(x)
rep
lac
es
all
fun
cti
on
s fi
de
term
inin
g X
wit
h t
he
co
ns
tan
t fu
nc
tio
nsX=x,
to
cre
ate
a m
uti
late
d m
od
elM
x
ST
RU
CT
UR
AL
MO
DE
LS
AN
D
INT
ER
VE
NT
ION
CA
US
AL
MO
DE
LS
AN
D
CO
UN
TE
RF
AC
TU
AL
S
De
fin
itio
n:
Th
e s
en
ten
ce
: “Y
wo
uld
be
y(i
n s
itu
ati
on
u),
ha
d X
be
enx,
”d
en
ote
d Y
x(u)
= y
, m
ea
ns
:
Th
e s
olu
tio
n f
or Y
in a
mu
tila
ted
mo
de
l M
x,
(i.e
., t
he
eq
ua
tio
ns
fo
r X
rep
lac
ed
by
X=
x)
wit
h i
np
ut U=u,
is e
qu
al
to y.
•
)(
),
()
(,
)(
:
uP
zZ
yY
Pz
uZ
yu
Yu
wx
wx
∑=
==
==
Jo
int
pro
ba
bili
tie
s o
f c
ou
nte
rfa
ctu
als
:
Th
e s
up
er-
dis
trib
uti
on
P*
is d
eri
ve
d f
rom
M.
Pa
rsim
on
ou
s,
co
ns
iste
nt,
an
d t
ran
sp
are
nt
AX
IOM
S O
F C
AU
SA
L
CO
UN
TE
RF
AC
TU
AL
S
1.
De
fin
ite
ne
ss
2.
Un
iqu
en
es
s
3.
Eff
ec
tiv
en
es
s
4.
Co
mp
os
itio
n
5.
Re
ve
rsib
ility
xu
Xt
sX
xy
=∈
∃)
( .
.
')'
)(
(&
))
((
xx
xu
Xx
uX
yy
=⇒
==
xu
Xxw
=)
(
)(
)(
)(
uY
uY
wu
Wx
xwx
=⇒
=
yu
Yw
uW
yu
Yx
xyxw
=⇒
==
)(
))
((
&)
((
:)
(y
uYx
=Y w
ou
ld b
ey, h
adX
be
enx
(in
sta
teU = u
)
GR
AP
HIC
AL
–C
OU
NT
ER
FA
CT
UA
LS
SY
MB
IOS
IS
Ev
ery
ca
us
al
gra
ph
ex
pre
ss
es
co
un
terf
ac
tua
ls
as
su
mp
tio
ns
, e
.g.,
X →
Y →
Z
co
ns
iste
nt,
an
d r
ea
da
ble
fro
m t
he
gra
ph
.
Ev
ery
th
eo
rem
in
SE
M i
s a
th
eo
rem
in
N-R
,
an
d c
on
ve
rse
ly.
)(
)(
,u
Yu
Yx
zx
=1
.M
iss
ing
arr
ow
s
Y←
Z
2.
Mis
sin
g a
rcs
YZ
yx
ZY
⊥⊥
ST
RU
CT
UR
AL
AN
AL
YS
IS:
SO
ME
US
EF
UL
RE
SU
LT
S
1.
Co
mp
lete
fo
rma
l s
em
an
tic
s o
f c
ou
nte
rfa
ctu
als
2.
Tra
ns
pa
ren
t la
ng
ua
ge
fo
r e
xp
res
sin
g a
ss
um
pti
on
s
3.
Co
mp
lete
so
luti
on
to
ca
us
al-
eff
ec
t id
en
tifi
ca
tio
n
4.
Le
ga
l re
sp
on
sib
ility
(b
ou
nd
s)
5.
No
n-c
om
plia
nc
e (
un
ive
rsa
l b
ou
nd
s)
6.
Inte
gra
tio
n o
f d
ata
fro
m d
ive
rse
so
urc
es
7.
Dir
ec
t a
nd
In
dir
ec
t e
ffe
cts
,
8.
Co
mp
lete
cri
teri
on
fo
r c
ou
nte
rfa
ctu
al
tes
tab
ility
RE
GR
ES
SIO
N V
S.
ST
RU
CT
UR
AL
EQ
UA
TIO
NS
(TH
E C
ON
FU
SIO
N O
F T
HE
CE
NT
UR
Y)
Re
gre
ss
ion
(c
laim
les
s):
Y = ax + a
1z 1+ a
2z 2+ ... + a
kzk+ ε
y
a ∂E
[Y|x,z]
/ ∂x
= R
yx⋅z
Y ε
y| X
,Z
Str
uc
tura
l (e
mp
iric
al,
fa
lsif
iab
le):
Y = bx+ b
1z 1+ b
2z 2+ ... + b
kzk+ u
y
b ∂E
[Y | do(x,z
)] /
∂x = ∂E
[Yx,z]
/∂x
As
su
mp
tio
ns
: Y
x,z=
Yx,z,w
cov(ui, uj) =
0 f
or
so
mei,j
⊥⊥=∆ =∆
RE
GR
ES
SIO
N V
S.
ST
RU
CT
UR
AL
EQ
UA
TIO
NS
(TH
E C
ON
FU
SIO
N O
F T
HE
CE
NT
UR
Y)
Re
gre
ss
ion
(c
laim
les
s):
Y = ax + a
1z 1+ a
2z 2+ ... + a
kzk+ ε
y
a ∂E
[Y|x,z]
/ ∂x
= R
yx⋅z
Y ε
y| X
,Z
Str
uc
tura
l (e
mp
iric
al,
fa
lsif
iab
le):
Y = bx+ b
1z 1+ b
2z 2+ ... + b
kzk+ u
y
b ∂E
[Y | do(x,z
)] /
∂x = ∂E
[Yx,z]
/∂x
As
su
mp
tio
ns
: Y
x,z=
Yx,z,w
cov(ui, uj) =
0 f
or
so
mei,j
⊥⊥=∆ =∆
Th
e m
oth
er
of
all
qu
es
tio
ns
:
“Wh
en
wo
uld
be
qu
al a
?,”
or
“Wh
at
kin
d o
f re
gre
ss
ors
sh
ou
ld Z
inc
lud
e?
”
An
sw
er:
Wh
en
Zs
ati
sfi
es
th
e b
ac
kd
oo
r c
rite
rio
n
Qu
es
tio
n:
Wh
en
is
be
sti
ma
ble
by
re
gre
ss
ion
me
tho
ds
?
An
sw
er:
gra
ph
ica
l c
rite
ria
av
aila
ble
TH
E B
AC
K-D
OO
R C
RIT
ER
ION
Gra
ph
ica
l te
st
of
ide
nti
fic
ati
on
P(y | do(x
))is
id
en
tifi
ab
le i
n G
if t
he
re i
s a
se
t Z
of
va
ria
ble
s s
uc
h t
ha
tZd
-se
pa
rate
s X
fro
m Y
inG
x.
Z6
Z3
Z2
Z5
Z1
XY
Z4
Z6
Z3
Z2
Z5
Z1
XY
Z4
Z
Mo
reo
ve
r, P
(y | do(x
))=
∑P
(y | x,z)
P(z
)
(“a
dju
sti
ng
” fo
r Z
)z
Gx
G
•do
-ca
lcu
lus
is
co
mp
lete
•C
om
ple
te g
rap
hic
al
cri
teri
on
fo
r id
en
tify
ing
ca
us
al
eff
ec
ts(S
hp
its
er
an
d P
ea
rl,
20
06
).
•C
om
ple
te g
rap
hic
al
cri
teri
on
fo
r e
mp
iric
al
tes
tab
ility
of
co
un
terf
ac
tua
ls
(Sh
pit
se
ra
nd
Pe
arl
, 2
00
7).
RE
CE
NT
R
ES
UL
TS
O
N
IDE
NT
IFIC
AT
ION
DE
TE
RM
ININ
G T
HE
CA
US
ES
OF
EF
FE
CT
S
(Th
e A
ttri
bu
tio
n P
rob
lem
)
•Y
ou
r H
on
or!
My
clie
nt
(Mr.
A)
die
d B
EC
AU
SE
he
us
ed
th
at
dru
g.
•
DE
TE
RM
ININ
G T
HE
CA
US
ES
OF
EF
FE
CT
S
(Th
e A
ttri
bu
tio
n P
rob
lem
)
•Y
ou
r H
on
or!
My
clie
nt
(Mr.
A)
die
d B
EC
AU
SE
he
us
ed
th
at
dru
g.
•C
ou
rt t
o d
ec
ide
if
it i
s M
OR
E P
RO
BA
BL
E T
HA
N
NO
Tth
at A
wo
uld
be
aliv
e B
UT
FO
Rth
e d
rug
!
P(?
| A
is d
ea
d,
too
k t
he
dru
g)
>0
.50
PN
=
TH
E P
RO
BL
EM
Se
ma
nti
ca
lP
rob
lem
:
1.
Wh
at
is t
he
me
an
ing
of PN
(x,y
):
“Pro
ba
bili
ty t
ha
t e
ve
nt y
wo
uld
no
t h
av
e o
cc
urr
ed
if
it w
ere
no
t fo
r e
ve
nt x,
giv
en
th
at x
an
d y
did
in
fa
ct
oc
cu
r.”
•
TH
E P
RO
BL
EM
Se
ma
nti
ca
lP
rob
lem
:
1.
Wh
at
is t
he
me
an
ing
of PN
(x,y
):
“Pro
ba
bili
ty t
ha
t e
ve
nt y
wo
uld
no
t h
av
e o
cc
urr
ed
if
it w
ere
no
t fo
r e
ve
nt x,
giv
en
th
at x
an
d y
did
in
fa
ct
oc
cu
r.”
An
sw
er:
Co
mp
uta
ble
fro
m M
),
|'(
),
('
yx
yY
Py
xPN
x=
=
TH
E P
RO
BL
EM
Se
ma
nti
ca
lP
rob
lem
:
1.
Wh
at
is t
he
me
an
ing
of PN
(x,y
):
“Pro
ba
bili
ty t
ha
t e
ve
nt y
wo
uld
no
t h
av
e o
cc
urr
ed
if
it w
ere
no
t fo
r e
ve
nt x,
giv
en
th
at x
an
d y
did
in
fa
ct
oc
cu
r.”
2.
Un
de
r w
ha
t c
on
dit
ion
ca
n PN
(x,y
)b
e l
ea
rne
d f
rom
sta
tis
tic
al
da
ta,
i.e
., o
bs
erv
ati
on
al,
ex
pe
rim
en
tal
an
d c
om
bin
ed
.
An
aly
tic
al
Pro
ble
m:
TY
PIC
AL
TH
EO
RE
MS
(Tia
na
nd
Pe
arl
, 2
00
0)
•B
ou
nd
s g
ive
n c
om
bin
ed
no
ne
xp
eri
me
nta
l a
nd
ex
pe
rim
en
tal
da
ta
≤≤
−
)(
)(
1
min
)(
)(
)(
0
max
x,y
P
y'P
PN
x,y
P
yP
yP
x'x'
)(
)(
)(
)(
)(
)(
x,y
P
yP
y|x'
P
y|x
P
y|x'
Py|x
PPN
x'−
+−
=
•Id
en
tifi
ab
ility
un
de
r m
on
oto
nic
ity
(C
om
bin
ed
da
ta)
co
rre
cte
d E
xc
es
s-R
isk
-Ra
tio
CA
N F
RE
QU
EN
CY
DA
TA
DE
CID
E
CA
N F
RE
QU
EN
CY
DA
TA
DE
CID
E
LE
GA
L R
ES
PO
NS
IBIL
ITY
?L
EG
AL
RE
SP
ON
SIB
ILIT
Y?
•N
on
ex
pe
rim
en
tal
da
ta:
dru
g u
sa
ge
pre
dic
ts l
on
ge
r lif
e
•E
xp
eri
me
nta
l d
ata
:d
rug
ha
s n
eg
ligib
le e
ffe
ct
on
su
rviv
al
Ex
pe
rim
en
tal
No
ne
xp
eri
me
nta
l
do(x
)do(x
′)x
x′D
ea
ths
(y)
16
14
22
8
Su
rviv
als
(y ′
)9
84
98
69
98
97
2
1,0
00
1,0
00
1,0
00
1,0
00
1.
He
ac
tua
lly d
ied
2.
He
us
ed
th
e d
rug
by
ch
oic
e
50
0.
),
|'(
'
>=
=∆y
xy
YP
PN
x
•C
ou
rt t
o d
ec
ide
(g
ive
n b
oth
da
ta):
Is i
t m
ore
pro
ba
ble
th
an
no
tth
at A
wo
uld
be
aliv
e
bu
t fo
rth
e d
rug
?
•P
lain
tiff
:M
r. A
is
sp
ec
ial.
SO
LU
TIO
N T
O T
HE
AT
TR
IBU
TIO
N P
RO
BL
EM
•W
ITH
PR
OB
AB
ILIT
Y O
NE
1 ≤
P(y
′ x′| x,y
) ≤
1
•C
om
bin
ed
da
ta t
ell
mo
re t
ha
t e
ac
h s
tud
y a
lon
e
CO
NC
LU
SIO
NS
Str
uc
tura
l-m
od
el
se
ma
nti
cs
, e
nri
ch
ed
wit
h l
og
ic
an
d g
rap
hs
, p
rov
ide
s:
•C
om
ple
te f
orm
al
ba
sis
fo
r th
e N-R
mo
de
l
•U
nif
ies
th
e g
rap
hic
al,
po
ten
tia
l-o
utc
om
e a
nd
str
uc
tura
l e
qu
ati
on
ap
pro
ac
he
s
•P
ow
erf
ul
an
d f
rie
nd
ly c
au
sa
l c
alc
ulu
s
(be
st
fea
ture
s o
f e
ac
h a
pp
roa
ch
)
![Bayesian Causal Inference - uni-muenchen.de...from causal inference have been attracting much interest recently. [HHH18] propose that causal [HHH18] propose that causal inference stands](https://img.dokumen.tips/doc/110x75/5ec457b21b32702dbe2c9d4c/bayesian-causal-inference-uni-from-causal-inference-have-been-attracting.jpg)
