1. Ambiguous Manipulations
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

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 don’t 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
1. don’t define causality - but will introduce axioms to connect probability to causality
2. many fields proceed without agreement on definition - probability, “force” in mechanics, interpretation of quantum mechanics, etc.
3. a number of different kinds of graphs represent probability distributions and independence - advantage of directed graphs is also represents causal relations
4. will introduce several extensions

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(X|Immediate 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
Post
Pre-experimental System
Room
Temperature
Wearing
Sweater

12. Modeling Ideal Interventions
Interventions on the Cause
Post
Pre-experimental System
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
Fat Hand - intervention - cholesterol drug -- arythmia

14. Calculating the Effect of Interventions
Pre-manipulation Joint Distribution
P(Exp,Inf,Rash) = P(Exp)P(Inf | Exp)P(Rash|Inf)
Intervention on Inf
Post-manipulation Joint Distribution
P(Exp,Inf,Rash) = P(Exp)P(Inf | I) P(Rash|Inf)

15. Causal Discovery from Observational Studies

16. Equivalence Class with Latents: PAGs: Partial Ancestral Graphs
Assumptions:
Acyclic graphs
Latent variables
Sample Selection Bias
Equivalence:
Independence over measured variables
1. represents set of conditional independence and distribution equivalent graphs
2. same adjacencies
3. undirected edges mean some contain edge one way, some contain other way
4. directed edge means they all go same way
5. Pearl and Verma -complete rules for generating from Meek, Andersson, Perlman, and Madigan, and Chickering
6. instance of chain graph
7. since data can’t distinguish, in absence of background knowledge is right output for search
8. what are they good for?

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

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 don’t 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 don’t know that we don’t 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
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations

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
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations

28. Logico-Causal Systems: Adding Edges
S: LDL, HDL, HD, D1, D2 S’: TC
System over S System over S U S’
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations
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’))]
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations
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’
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations
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
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations

32. 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 HD
Can’t tell TC
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations

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.
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations
Example:
If observed system has no deterministic relations
All orientations due to marginal independence relations are still valid

34. 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
0. Pearl calls stability
1. all conditional independencies that hold entailed by Markov assumption
2. a kind of simplicity assumption
3. graph it is faithful to has more degrees of freedom than other graphs that fit distribution
4. violation is zero Lebesgue measure
5. on a variety of scoring rules, faithful graph does best in limit
6. doesn’t say anthing about “almost” violations

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,
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 1