Recent Advanced in Causal Modelling Using Directed Graphs

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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
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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(No Transcript)
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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
  dont 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

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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

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Causal Bayes Networks
The Joint Distribution Factors According to the
Causal Graph, i.e., for all X in V P(V)
?P(XImmediate Causes of(X))

• P(S 0) .7
• P(S 1) .3
• P(YF 0 S 0) .99 P(LC 0 S 0) .95
• P(YF 1 S 0) .01 P(LC 1 S 0) .05
• P(YF 0 S 1) .20 P(LC 0 S 1) .80
• P(YF 1 S 1) .80 P(LC 1 S 1) .20

P(S,Y,F) P(S) P(YF S) P(LC S)
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Modeling Ideal Interventions
Interventions on the Effect
Pre-experimental System
Room Temperature
Wearing Sweater
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Modeling Ideal Interventions
Interventions on the Cause
Pre-experimental System
Post
Room Temperature
Wearing Sweater
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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 Post-intervention graph?
Pre-intervention graph
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Calculating the Effect of Interventions
Pre-manipulation Joint Distribution
P(Exp,Inf,Rash) P(Exp)P(Inf Exp)P(RashInf)
Intervention on Inf
Post-manipulation Joint Distribution
P(Exp,Inf,Rash) P(Exp)P(Inf I) P(RashInf)
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Causal Discovery from Observational Studies
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Equivalence Class with LatentsPAGs Partial
Ancestral Graphs

• 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
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3. The Ambiguity of Manipulation

• Assumptions
• Causal graph known
  (Cholesterol is a cause of Heart Condition)
• No Unmeasured Common Causes

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
dont act uniformly, E.g., Total Blood
Cholesterol (TC) 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

• 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 dont know that we dont
  know.

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Examples Abound
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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

• 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
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 LDL, HDL, HD, D1, D2 S TC System over
S System over S U S
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

• 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
Still sound but less informative
Observed System TC, HD, D1, D2
Manipulate Effect on Assume manipulation unambiguous Manipulation Maybe ambiguous
Disease 1 Disease 2 None None
Disease 1 HD Cant tell Cant tell
Disease 1 TC Cant tell Cant tell
Disease 2 Disease 1 None None
Disease 2 HD Cant tell Cant tell
Disease 2 TC Cant tell Cant tell
TC Disease 1 None Cant tell
TC Disease 2 None Cant tell
TC HD Cant tell Cant tell
HD Disease 1 None Cant tell
HD Disease 2 None Cant tell
HD TC Cant tell Cant tell
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Effect on Prediction Algorithm
Observed System TC, HD, D1, D2, 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