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Causal Inference and

Ambiguous Manipulations

Richard Scheines

Grant Reaber, Peter SpirtesCarnegie Mellon University

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1. MotivationWanted: 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 Interventions3. 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

WearingSweater

Room

Temperature

Pre-experimental SystemPost

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Modeling Ideal Interventions

Interventions on the Cause

Pre-experimental SystemPost

WearingSweater

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 EdgesS: 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