Embed Size (px)
DESCRIPTION
THE OVERRIDING THEME. Define Q ( M ) as a counterfactual expression Determine conditions for the reduction If reduction is feasible, Q is inferable. Demonstrated on three types of queries:. Q 1 : P ( y | do ( x )) Causal Effect (= P ( Y x =y ) ) - PowerPoint PPT Presentation
Citation preview
THE OVERRIDING THEME
1. Define Q(M) as a counterfactual expression2. Determine conditions for the reduction
3. If reduction is feasible, Q is inferable.
• Demonstrated on three types of queries:
)()()()( exp MPMQMPMQ or
Q1: P(y|do(x)) Causal Effect (= P(Yx=y))Q2: P(Yx = y | x, y) Probability of necessityQ3: Direct Effect)(
'xZxYE
• Modeling: Statistical vs. Causal• Causal Models and Identifiability• Inference to three types of claims:
1. Effects of potential interventions
2. Claims about attribution (responsibility)
3. Claims about direct and indirect effects• Actual Causation and Explanation• Robustness of Causal Claims
OUTLINE
ROBUSTNESS:MOTIVATION
Smokingx y
Genetic Factors (unobserved)
Cancer
u
In linear systems: y = x + u cov (x,u) = 0 is identifiable. = Ryx
ROBUSTNESS:MOTIVATION
The claim Ryx is sensitive to the assumption cov (x,u) = 0.
Smokingx
Genetic Factors (unobserved)
Cancer
is non-identifiable if cov (x,u) ≠ 0.
y
u
ROBUSTNESS:MOTIVATION
Z – Instrumental variable; cov(z,u) = 0
Smokingy
Genetic Factors (unobserved)
Cancer
u
x
ZPrice ofCigarettes
xz
yz
xz
yzRR
RR
is identifiable, even if cov (x,u) ≠ 0
ROBUSTNESS:MOTIVATION
Smokingy
Genetic Factors (unobserved)
Cancer
u
x
ZPrice ofCigarettes
10
10
Suppose
xz
yzyx R
RR
Claim “ = Ryx” is likely to be true
Smoking
ROBUSTNESS:MOTIVATION
Z1
Price ofCigarettes
Invoking several instruments
If =1 = 2, claim “ = 0” is more likely correct2
22
1
10
xz
yz
xz
yzyx R
RRR
R 1
x y
Genetic Factors (unobserved)
Cancer
u
PeerPressure
Z2
ROBUSTNESS:MOTIVATION
Z1
Price ofCigarettes
x y
Genetic Factors (unobserved)
Cancer
u
PeerPressure
Z2
Smoking
Greater surprise: 1 = 2 = 3….= n = qClaim = q is highly likely to be correct
Z3
Zn
Anti-smoking Legislation
ROBUSTNESS:MOTIVATION
Assume we have several independent estimands of , and
x y
Given a parameter in a general graph
Find the degree to which is robust to violations of model assumptions
1 = 2 = …n
ROBUSTNESS:ATTEMPTED FORMULATION
Bad attempt: Parameter is robust (over identifies)
f1, f2: Two distinct functions
)()( 21 ff
distinct. are
then constraint induces model if
)]([)]([)()]([)(
,0)(
21
gtgtfgtf
g
i
if:
ROBUSTNESS:MOTIVATION
x y
Genetic Factors (unobserved)
Cancer
u
Smoking
Is robust if 0 = 1?
sxsysysx
yx
RRRR
R
,1
0
s
Symptom
ROBUSTNESS:MOTIVATION
x y
Genetic Factors (unobserved)
Cancer
u
Smoking
Symptoms do not act as instruments
remains non-identifiable if cov (x,u) ≠ 0
s
Symptom
Why? Taking a noisy measurement (s) of an observed variable (y) cannot add new information
ROBUSTNESS:MOTIVATION
x
Genetic Factors (unobserved)
Cancer
u
Smoking
Adding many symptoms does not help.
remains non-identifiable
ySymptom
S1
S2
Sn
INDEPENDENT:BASED ON DISTINCT SETS OF ASSUMPTION
u
z yx
u
zyx
EstimandEstimand AssumptiomsAssumptioms
xz
yz
yx
RR
R
1
0
others
0),cov( ux
EstimandEstimand AssumptiomsAssumptioms
zyzx
yx
RR
R
1
0
others
0),cov(
0),cov(
ux
ux
RELEVANCE:FORMULATION
Definition 8 Let A be an assumption embodied in model M, and p a parameter in M. A is said to be relevant to p if and only if there exists a set of assumptions S in M such that S and A sustain the identification of p but S alone does not sustain such identification.
Theorem 2 An assumption A is relevant to p if and only if A is a member of a minimal set of assumptions sufficient for identifying p.
ROBUSTNESS:FORMULATION
Definition 5 (Degree of over-identification)A parameter p (of model M) is identified to degree k (read: k-identified) if there are k minimal sets of assumptions each yielding a distinct estimand of p.
ROBUSTNESS:FORMULATION
x yb
zc
Minimal assumption sets for c.
x y zc x y zc
G3G2
x y zc
G1
Minimal assumption sets for b. x yb
z
FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS
FROM PARAMETERS TO CLAIMS
DefinitionA claim C is identified to degree k in model M (graph G), if there are k edge supergraphs of G that permit the identification of C, each yielding a distinct estimand.
TE(x,z) = Rzx TE(x,z) = Rzx Rzy ·x
x y zx y z
e.g., Claim: (Total effect) TE(x,z) = q x y z
FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS
FROM PARAMETERS TO CLAIMS
DefinitionA claim C is identified to degree k in model M (graph G), if there are k edge supergraphs of G that permit the identification of C, each yielding a distinct estimand.
x y zx y z
e.g., Claim: (Total effect) TE(x,z) = q x y z
Nonparametric y x
xPyxzPxyPxzTExzPzxTE'
)'(),'|()|(),()|(),(
SUMMARY OF ROBUSTNESS RESULTS
1. Formal definition to ROBUSTNESS of causal claims:“A claim is robust when it is insensitive to
violations of some of the model assumptions relevant to substantiating that claim.”
2. Graphical criteria and algorithms for computing the degree of robustness of a given causal claim.
CONCLUSIONS
Structural-model semantics enriched with logic + graphs leads to formal interpretation and practical assessments of wide variety of causal and counterfactual relationships.
