Transcript Recent Advanced in Causal Modelling Using Directed Graphs

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
Conditioning is not the same as intervening
P(Lung Cancer | Tar-stained teeth = no)

P(Lung Cancer | Tar-stained teeth set= no)
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)
Individual
Day Care
John
A lot
Mary
None
- Response (Y)
Aggressivenes
s
A lot
A little
- 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
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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
The Joint Distribution Factors
Smoking [0,1]
According to the Causal Graph,
Yellow Fingers
[0,1]
Lung Cancer
[0,1]
P(S = 0) = .7
P(S = 1) = .3
P(YF = 0 | S = 0) = .99
P(YF = 1 | S = 0) = .01
P(YF = 0 | S = 1) = .20
P(YF = 1 | S = 1) = .80
i.e., for all X in V
P(V) = P(X|Immediate
Causes of(X))
P(S,Y,F) = P(S) P(YF | S) P(LC | S)
P(LC = 0 | S = 0) = .95
P(LC = 1 | S = 0) = .05
P(LC = 0 | S = 1) = .80
P(LC = 1 | S = 1) = .20
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Modeling Ideal Interventions
Interventions on the Effect
Post
Pre-experimental System
Wearing
Sweater
Room
Temperature
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Modeling Ideal Interventions
Interventions on the Cause
Post
Pre-experimental System
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
Intervene to change Inf
Pre-intervention graph
Post-intervention graph?
Exp
Exp
Inf
Inf
Rash
Rash
I
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Calculating the Effect of
Interventions
Exp
Inf
Rash
Pre-manipulation Joint Distribution
P(Exp,Inf,Rash) = P(Exp)P(Inf | Exp)P(Rash|Inf)
Intervention on Inf
Exp
Inf
I
Rash
Post-manipulation Joint Distribution
P(Exp,Inf,Rash) = P(Exp)P(Inf | I) P(Rash|Inf)
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Causal Discovery from
Observational Studies
Equivalence Class of
Causal Graphs
X1
X1
X1
X2
X2
X2
X3
X3
X3
Causal Markov Axiom
(D-separation)
Independence
Relations
Discovery Algorithm
X1
X 3 | X2
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Equivalence Class with Latents:
PAGs: Partial Ancestral Graphs
Assumptions:
X1
PAG
• Acyclic graphs
X3
• Latent variables
Represents
• Sample Selection Bias
X1
Equivalence:
X2
X1
X3
• Independence over
measured variables
X2
X2
T1
X3
etc.
X1
X2
X1
T1
X3
T1
X2
X3
T2
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Causal Inference from
Observational Studies
Knowing when we know enough to
calculate the effect of Interventions
The Prediction Algorithm (SGS, 2000)
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Causal Discovery from
Observational Studies
Equivalence Class (PAG)
X1
X2
X3
Prediction Algorithm
X4
Discovery Algorithm
Observed Independence
X1 _||_ X4
X1 _||_ X3 | X2
X4 _||_ X3 | X2
Predictions?
P(X3 | X2set)
yes
P(X2 | X1set)
Don’t know
P(X1 | X2set)
yes
….
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3. The Ambiguity of Manipulation
Assumptions
• Causal graph known
Total Blood
Cholesterol
(Cholesterol is a cause of Heart Condition)
• No Unmeasured Common Causes
Heart
Disease
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
Total Blood Cholesterol =
HDL
+
+
LDL
Heart
Disease
•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
Total Blood Cholesterol =
HDL
+
+
LDL
Heart
Disease
• 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
Total TV =
Violent Junk
+
PBS, Discovery Channel
Total Day Care =
Overcrowded, Poor Quality
+
High Quality
Social
Adjustment
+
+
-
Aggressiveness
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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
HDL
TC
LDL
?
Heart
Disease
• 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’: TC
S: LDL, HDL, HD, D1, D2
System over S U S’
System over S
D1
D2
D1
HDL
LDL
D2
HDL
LDL
TC
?
HD
HD
TC  HD iff manipulations of TC are unambiguous wrt HD
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Logico-Causal Systems:
Unambiguous Manipulations
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’))]
TC  HD iff all manipulations of TC are unambiguous wrt HD
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Logico-Causal Systems: Removing Edges
S’: TC
S: LDL, HDL, HD, D1, D2
System over S U S’
System over S
D1
D2
D1
HDL
LDL
D2
HDL
LDL
TC
?
?
HD
HD
Remove LDL  HD iff LDL _||_ HD | TC
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Logico-Causal Systems: Faithfulness
Faithfulness: Independences entailed by structure, not by
special parameter values. Crucial to inference
D1
D2
Effect of TC on HD unambiguous
HDL
LDL
Unfaithfulness: LDL _||_ HDL | TC
TC
HD
Because LDL and TC determine HDL, and
similarly, HDL and TC determine TC
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Effect on Prediction Algorithm
Observed System:
Still sound – but less informative
Manipulate:
Effect on:
Assume
manipulation
unambiguous
Manipulation
Maybe
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
TC, HD, D1, D2
D1
D2
HDL
LDL
TC
?
?
HD
?
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Effect on Prediction Algorithm
Observed System:
TC, HD, D1, D2, X
X
D1
D2
Not completely sound
No general characterization
of when the Prediction
algorithm, suitably modified,
is still informative and sound.
Conjectures, but no proof yet.
HDL
LDL
TC
?
HD
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 of
Ambiguous 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
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