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Causal Inference and
Ambiguous Manipulations
Richard Scheines
Grant Reaber, Peter Spirtes
Carnegie Mellon University
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1. Motivation
Wanted: Answers to Causal Questions:
• Does attending Day Care cause Aggression?
• Does watching TV cause obesity?
• How can we answer these questions
empirically?
• When and how can we estimate the size of
the effect?
• Can we know our estimates are reliable?
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Causation & Intervention
P(Lung Cancer | Tar-stained teeth = no)
P(Lung Cancer | Tar-stained teeth set= no)
Conditioning is not the same as intervening
Show Teeth Slides
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Gender
CEO Earings
Gender
CEO Earings
I
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Causal Inference: Experiments
Gold Standard: Randomized Clinical Trials
- Intervene: Randomly assign treatment
- Observe Response
Estimate P( Response | Treatment assigned)
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Causal Inference: Observational Studies
Collect a sample on
- Potential Causes (X)
- Response (Y)
- Covariates (potential confounders Z)
Estimate P(Y | X, Z)
• Highly unreliable
• We can estimate sampling variability, but we don’t know
how to estimate specification uncertainty from data
Individual Day Care Aggressiveness
John
Mary
A lot
None
A lot
A little
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2. Progress 1985 – Present
1. Representing causal structure, and
connecting it to probability
2. Modeling Interventions
3. Indistinguishability and Discovery
Algorithms
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Representing Causal Structures
Causal Graph G = {V,E} Each edge X Y represents a direct causal claim:
X is a direct cause of Y relative to V
Exposure Infection Symptoms
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Direct Causation
X is a direct cause of Y relative to S, iff z,x1 x2 P(Y | X set= x1 , Z set= z)
P(Y | X set= x2 , Z set= z)
where Z = S - {X,Y} X Y
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Causal Bayes Networks
P(S = 0) = .7P(S = 1) = .3
P(YF = 0 | S = 0) = .99 P(LC = 0 | S = 0) = .95P(YF = 1 | S = 0) = .01 P(LC = 1 | S = 0) = .05P(YF = 0 | S = 1) = .20 P(LC = 0 | S = 1) = .80P(YF = 1 | S = 1) = .80 P(LC = 1 | S = 1) = .20
Smoking [0,1]
Lung Cancer[0,1]
Yellow Fingers[0,1]
P(S,Y,F) = P(S) P(YF | S) P(LC | S)
The Joint Distribution Factors
According to the Causal Graph,
i.e., for all X in V
P(V) = P(X|Immediate Causes of(X))
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Modeling Ideal Interventions
Interventions on the Effect
Wearing
Sweater
Room
Temperature
Pre-experimental SystemPost
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Modeling Ideal Interventions
Interventions on the Cause
Pre-experimental SystemPost
Wearing
Sweater
Room
Temperature
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Interventions & Causal Graphs
• Model an ideal intervention by adding an “intervention” variable outside the original system
• Erase all arrows pointing into the variable intervened upon
Exp Inf
Rash
Intervene to change Inf
Post-intervention graph?Pre-intervention graph
Exp Inf Rash
I
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Calculating the Effect of Interventions
Pre-manipulation Joint Distribution
P(Exp,Inf,Rash) = P(Exp)P(Inf | Exp)P(Rash|Inf)
Intervention on Inf
Exp Inf
Rash
Post-manipulation Joint Distribution
P(Exp,Inf,Rash) = P(Exp)P(Inf | I) P(Rash|Inf)
Exp Inf
Rash
I
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Causal Discovery from Observational Studies
X3 | X2 X1
X2 X3 X1
Causal Markov Axiom(D-separation)
IndependenceRelations
Equivalence Class ofCausal Graphs
X2 X3 X1
X2 X3 X1
Discovery Algorithm
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Equivalence Class with Latents:PAGs: Partial Ancestral Graphs
X2
X3
X1
X2
X3
Represents
PAG
X1 X2
X3
X1
X2
X3
T1
X1
X2
X3
X1
etc.
T1
T1 T2
Assumptions:
• Acyclic graphs
• Latent variables
• Sample Selection Bias
Equivalence:
• Independence over measured variables
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Knowing when we know enough to calculate the effect of Interventions
The Prediction Algorithm (SGS, 2000)
Causal Inference from
Observational Studies
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Causal Discovery from Observational Studies
X2 X3 X1 Prediction Algorithm
Equivalence Class (PAG)
X4
Predictions? P(X3 | X2set) yes P(X2 | X1set) Don’t know P(X1 | X2set) yes ….
Observed Independence
X1 _||_ X4 X1 _||_ X3 | X2
X4 _||_ X3 | X2
Discovery Algorithm
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3. The Ambiguity of Manipulation
Assumptions
• Causal graph known (Cholesterol is a cause of Heart Condition)
• No Unmeasured Common Causes
Heart Disease
Total Blood Cholesterol
Therefore
The manipulated and unmanipulated distributions are the same:
P(H | TC = x) = P(H | TC set= x)
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The Problem with Predicting the Effects of Acting
Problem – the cause is a composite of causes that don’t act uniformly,
E.g., Total Blood Cholesterol (TC) = HDL + LDL
Heart Disease
Total Blood Cholesterol = HDL
+ LDL +
-
•The observed distribution over TC is determined by the unobserved joint distribution over HDL and LDL
• Ideally Intervening on TC does not determine a joint distribution for HDL and LDL
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The Problem with Predicting the Effects of Setting TC
Heart Disease
Total Blood Cholesterol = HDL
+ LDL +
-
• P(H | TC set1= x) puts NO constraints on P(H | TC set2= x),
• P(H | TC = x) puts NO constraints on P(H | TC set= x)
• Nothing in the data tips us off about our ignorance, i.e., we don’t
know that we don’t know.
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Examples Abound
Social Adjustment
Total TV = Violent Junk
+ PBS, Discovery Channel
+
-
Aggressiveness Total Day Care =
Overcrowded, Poor Quality +
High Quality
+ -
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Possible Ways Out
• Causal Graph is Not Known:
Cholesterol does not really cause Heart Condition
• Confounders (unmeasured common causes) are present:
LDL and HDL are confounders
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Cholesterol is not really a cause of Heart Condition
Relative to a set of variables S (and a background),
X is a cause of Y iff x1 x2 P(Y | X set= x1) P(Y | X set= x2)
• Total Cholesterol is a cause of Heart Disease
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Cholesterol is not really a cause of Heart Condition
Is Total Cholesterol is a direct cause of Heart Condition relative to: {TC, LDL, HDL, HD}?
• TC is logically related to LDL, HDL, so manipulating it once LDL and HDL are set is impossible.
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LDL, HDL are confounders
Heart Disease TC
HDL LDL
?
• No way to manipulate TCl without affecting HDL, LDL
• HDL, LDL are logically related to TC
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Logico-Causal Systems
S: Atomic Variables
• independently manipulable
• effects of all manipulations are unambiguous
S’: Defined Variables
• defined logically from variables in S
For example:
S: LDL, HDL, HD, Disease1, Disease2
S’: TC
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Logico-Causal Systems: Adding Edges
S: LDL, HDL, HD, D1, D2 S’: TC
System over S System over S U S’ D1 D2
LDL HDL
HD
D1 D2
LDL HDL
HD
TC
?
TC HD iff manipulations of TC are unambiguous wrt HD
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Logico-Causal Systems: Unambiguous Manipulations
TC HD iff all manipulations of TC are unambiguous wrt HD
For each variable X in S’, let Parents(X’) be the set of variables in S that logically determine X’, i.e.,
X’ = f(Parents(X’)), e.g., TC = LDL + HDL
Inv(x’) = set of all values p of Parents(X’) s.t., f(p) = x’
A manipulation of a variable X’ in S’ to a value x’
wrt another variable Y is unambiguous iff
p1≠ p2 [P(Y | p1 Inv(x’)) = P(Y | p2 Inv(x’))]
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Logico-Causal Systems: Removing Edges
S: LDL, HDL, HD, D1, D2 S’: TC
System over S System over S U S’ D1 D2
LDL HDL
HD
D1 D2
LDL HDL
HD
TC
? ?
Remove LDL HD iff LDL _||_ HD | TC
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Logico-Causal Systems: Faithfulness
D1 D2
LDL HDL
HD
TC
Faithfulness: Independences entailed by structure, not by
special parameter values. Crucial to inference
Effect of TC on HD unambiguous
Unfaithfulness: LDL _||_ HDL | TC
Because LDL and TC determine HDL, and
similarly, HDL and TC determine TC
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Effect on Prediction Algorithm
Manipulate: Effect on: Assume manipulation unambiguous
ManipulationMaybe ambiguous
Disease 1 Disease 2 None None
Disease 1 HD Can’t tell Can’t tell
Disease 1 TC Can’t tell Can’t tell
Disease 2 Disease 1 None None
Disease 2 HD Can’t tell Can’t tell
Disease 2 TC Can’t tell Can’t tell
TC Disease 1 None Can’t tell
TC Disease 2 None Can’t tell
TC HD Can’t tell Can’t tell
HD Disease 1 None Can’t tell
HD Disease 2 None Can’t tell
HD TC Can’t tell Can’t tell
Observed System:
TC, HD, D1, D2 D1 D2
LDL HDL
HD
TC
? ? ?
Still sound – but less informative
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Effect on Prediction Algorithm
Observed System:
TC, HD, D1, D2, X
D1 D2
LDL HDL
HD
TC
?
X
Not completely sound
No general characterization of when the Prediction algorithm, suitably modified, is still informative and sound. Conjectures, but no proof yet.
Example:• If observed system has no deterministic relations• All orientations due to marginal independence relations are still valid
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Effect on Causal Inference ofAmbiguous Manipulations
Experiments, e.g., RCTs:
Manipulating treatment is• unambiguous sound• ambiguous unsound
Observational Studies, e.g., Prediction Algorithm:
Manipulation is• unambiguous potentially sound• ambiguous potentially sound
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References
• Causation, Prediction, and Search, 2nd Edition, (2000), by P. Spirtes, C. Glymour, and R. Scheines ( MIT Press)
• Causality: Models, Reasoning, and Inference, (2000), Judea Pearl, Cambridge Univ. Press
• Spirtes, P., Scheines, R.,Glymour, C., Richardson, T., and Meek, C. (2004), “Causal Inference,” in Handbook of Quantitative Methodology in the Social Sciences, ed. David Kaplan, Sage Publications, 447-478
• Spirtes, P., and Scheines, R. (2004). Causal Inference of Ambiguous Manipulations. in Proceedings of the Philosophy of Science Association Meetings, 2002.
• Reaber, Grant (2005). The Theory of Ambiguous Manipulations. Masters Thesis, Department of Philosophy, Carnegie Mellon University