chen avin ilya shpitser judea pearl computer science department ucla
DESCRIPTION
Chen Avin Ilya Shpitser Judea Pearl Computer Science Department UCLA. IDENTIFIABILITY OF PATH-SPECIFIC EFFECTS. QUESTIONS ASKED. Why path-specific effects? What are the semantics of path-specific effects (in nonlinear and nonparametric models)? - PowerPoint PPT PresentationTRANSCRIPT
Chen Avin
Ilya Shpitser
Judea Pearl
Computer Science Department
UCLA
IDENTIFIABILITY OF PATH-SPECIFIC
EFFECTS
QUESTIONS ASKED
• Why path-specific effects?• What are the semantics of path-specific effects
(in nonlinear and nonparametric models)?• What are the policy implications of path-specific
effects?• When can path-specific effects be estimated
consistently from experimental or nonexperimental data?
• Can these conditions be verified from accessible causal knowledge, i.e., graphs?
1. Direct (or indirect) effect may be more transportable.2. Indirect effects may be prevented or controlled.
3. Direct (or indirect) effect may be forbidden
WHY DECOMPOSEEFFECTS?
Pill
Thrombosis
Pregnancy
+
+
Gender
Hiring
Qualification
EFFECT-DECOMPOSITIONIN LINEAR MODELS
X Z
Y
ca
b
effect Indirect effect Direct effect Total
a bc
Definition: ))(),(|( zxYEx
a do do
CAUSAL MODELS AND COUNTERFACTUALS
Definition: A causal model is a 3-tupleM = V,U,F
(i) V = {V1…,Vn} endogenous variables,(ii) U = {U1,…,Um} background variables (unit)(iii) F = set of n functions,
The sentence: “Y would be y (in unit u), had X been x,”denoted Yx(u) = y, is the solution for Y in a mutilated model Mx, with the equations for X replaced by X = x. (“unit-based potential outcome”)
),( uvfv ii
COUNTERFACTUALS:STRUCTURAL SEMANTICS
Notation: Yx(u) = y Y has the value y in the solution to a mutilated system of equations, where the equation for X is replaced by a constant X=x.
u
Yx(u)=y
Z
W
X=x
u
Y
Z
W
X
Probability of Counterfactuals:
FunctionalBayes Net
))(|()())(()( xdoyPu
uPyuxYPyxYP
)(uM,P
tindependen- ))(),(|(
))(|(
DETEIE
ZzdoxdoYEx
DE
xdoYEx
TE
TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE CONTROLLED-BASED
SEMANTICS IN LINEAR MODELS
X Z
Y
ca
b z = bx + 1
y = ax + cz + 2
a + bc
bc
a
z = f (x, 1)y = g (x, z, 2)
????
))(),(|(
))(|(
IE
zdoxdoYEx
DE
xdoYEx
TE
X Z
Y
CONTROLLED-BASED SEMANTICS NONTRIVIAL IN NONLINEAR MODELS(even when the model is completely specified)
Dependent on z?
Void of operational meaning?
``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’
[Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))]
x = male, x = femaley = hire, y = not hirez = applicant’s qualifications
LEGAL DEFINITIONS OF DIRECT EFFECT
(FORMALIZING DISCRIMINATION)
NO DIRECT EFFECT
',' ' xxxx YYYYxZxZ
z = f (x, u)y = g (x, z, u)
X Z
Y
NATURAL SEMANTICS OFAVERAGE DIRECT EFFECTS
Average Direct Effect of X on Y:The expected change in Y, when we change X from x0 to x1 and, for each u, we keep Z constant at whatever value it attained before the change.
In linear models, DE = Controlled Direct Effect
][001 xZx YYE
x
);,( 10 YxxDE
Robins and Greenland (1992) – “Pure”
POLICY IMPLICATIONS(Who cares?)
f
GENDER QUALIFICATION
HIRING
What is the direct effect of X on Y?
Is employer guilty of sex-discrimination given data on (X,Y,Z)?
X Z
Y
CAN WE IGNORE THIS LINK?
tYYE xZxx
][0
01
z = f (x, u)y = g (x, z, u)
X Z
Y
NATURAL SEMANTICS OFINDIRECT EFFECTS
Indirect Effect of X on Y:The expected change in Y when we keep X constant, say at x0, and let Z change to whatever value it would have attained had X changed to x1.
In linear models, IE = TE - DE
][010 xZx YYE
x
);,( 10 YxxIE
POLICY IMPLICATIONS(Who cares?)
f
GENDER QUALIFICATION
HIRING
What is the indirect effect of X on Y?
The effect of Gender on Hiring if sex discriminationis eliminated.
X Z
Y
IGNORE
SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS
Consider the quantity
Given M, P(u), Q is well defined
Given u, Zx*(u) is the solution for Z in Mx*, call it z
is the solution for Y in Mxz
Can Q be estimated from data?
Experimental: nest-free expressionNonexperimental: subscript-free expression
)]([ )(*uYEQ uxZxu
entalnonexperim
alexperiment
)()(*uY uxZx
Corollary 3:The average direct effect in Markovian models is identifiable from nonexperimental data, and it is given by
where S stands for any sufficient set of covariates.
IDENTIFICATION INMARKOVIAN MODELS
X ZExample:S =
Y
s z
sPsxzPzxYEzxYEYxxDE )()*,|()*,|(),|()*;,(
z
xzPzxyEzxYEYxxDE *)|()*,|(),|()*;,(
Y
Z
X
W
x*
z* = Zx* (u)
GENERAL PATH-SPECIFICEFFECTS (Def.)
)),(*),(();,(* ugpagpafgupaf iiiii
*);,();,( **gMMg YxxTEYxxE
Y
Z
X
W
Form a new model, , specific to active subgraph g*gM
Definition: g-specific effect
EFFECT-INVARIANT
Rule 1 Rule 2
MAIN RESULT
Applying the two rules results in one of two cases:
Case 1: we obtain a ‘kite pattern.’ Then the path-specific effect is not identifiable.
R - Recanting witness
Z
Y
MAIN RESULT (Cont.)
X
Y
ZW Z’ Z”
Case 2: all blocked edges emanate from the root node. Then the effect is identifiable.
AZT EXAMPLE REVISITED
AZT
Pneumonia
Antibiotics
Headaches
Painkillers
Survival
Painkiller contribution to the total effect of AZT on survival
Antibiotics
AZT
PneumoniaHeadaches
Painkillers
Survival
Antibiotics contribution to the total effect of AZT on survival
RECANTING WITNESS
Antibiotics
AZT
PneumoniaHeadaches
Painkillers
Survival
Antibiotics contribution to the total effect of AZT on survival
R-Recanting Witness
R behaves as I
R behaves as II
P(RX,RX*) is not experimentally identifiable
SUMMARY OF RESULTS
1. Formal semantics of path-specific effects, based on signal blocking, instead of value fixing.
2. Path-analytic techniques extended to nonlinear and nonparametric models.
3. Meaningful (graphical) conditions for estimating effects from experimental and nonexperimental data.
4. Graphical techniques of inferring effects of policies involving signal blocking.