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Recent Advanced in Causal Modelling Using Directed Graphs

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Ambiguous Manipulations. Richard Scheines. Grant Reaber, Peter ... ambiguous potentially sound. 35. References ... Causal Inference of Ambiguous Manipulations. ... – PowerPoint PPT presentation

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Title: Recent Advanced in Causal Modelling Using Directed Graphs


1
Causal Inference and Ambiguous Manipulations
Richard Scheines Grant Reaber, Peter
Spirtes Carnegie Mellon University
2
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?

3
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
4
(No Transcript)
5
Causal Inference Experiments
  • Gold Standard Randomized Clinical Trials -
    Intervene Randomly assign treatment - Observe
    Response
  • Estimate P( Response Treatment assigned)

6
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

7
2. Progress 1985 Present
  • Representing causal structure, and connecting it
    to probability
  • Modeling Interventions
  • Indistinguishability and Discovery Algorithms

8
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

9
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

10
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)
11
Modeling Ideal Interventions
Interventions on the Effect
Pre-experimental System
Room Temperature
Wearing Sweater
12
Modeling Ideal Interventions
Interventions on the Cause
Pre-experimental System
Post
Room Temperature
Wearing Sweater
13
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
14
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)
15
Causal Discovery from Observational Studies
16
Equivalence Class with LatentsPAGs Partial
Ancestral Graphs
  • Assumptions
  • Acyclic graphs
  • Latent variables
  • Sample Selection Bias
  • Equivalence
  • Independence over measured variables

17
Knowing when we know enough to calculate the
effect of Interventions The Prediction
Algorithm (SGS, 2000)
Causal Inference from Observational Studies
18
Causal Discovery from Observational Studies
19
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)
20
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

21
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.

22
Examples Abound
23
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

24
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

25
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.

26
LDL, HDL are confounders
  • No way to manipulate TCl without affecting HDL,
    LDL
  • HDL, LDL are logically related to TC

27
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

28
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
29
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
30
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
31
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

32
Effect on Prediction Algorithm
Still sound but less informative
Observed System TC, HD, D1, D2
33
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

34
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

35
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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