1. Department of Computer Science
REASONING WITH
CAUSE AND EFFECT
Judea Pearl
Department of Computer Science
UCLA

2. OUTLINE Modeling: Statistical vs. Causal
Causal Models and Identifiability
Inference to three types of claims:
Effects of potential interventions
Claims about attribution (responsibility)
Claims about direct and indirect effects
Actual Causation and Explanation
Falsifiability and Corroboration

3. TRADITIONAL STATISTICAL
INFERENCE PARADIGM P
Joint
Distribution
Q(P)
(Aspects of P)
Data
Inference
e.g.,
Infer whether customers who bought product A
would also buy product B.
Q = P(B|A)

4. THE CAUSAL INFERENCE PARADIGM M Data-generating Model Q(M)
(Aspects of M)
Data
Inference
Some Q(M) cannot be inferred from P.
e.g.,
Infer whether customers who bought product A
would still buy A if we double the price.

5. FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
Data
joint
distribution
inferences
from passive
observations
Probability and statistics deal with static relations
Probability
Statistics

6. FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
Probability and statistics deal with static relations
Statistics
Probability
inferences
from passive
observations
joint
distribution
Data
Causal analysis deals with changes (dynamics)
i.e. What remains invariant when P changes.
P does not tell us how it ought to change
e.g. Curing symptoms vs. curing diseases
e.g. Analogy: mechanical deformation

7. FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
Data
joint
distribution
inferences
from passive
observations
Probability and statistics deal with static relations
Probability
Statistics
Causal
Model
Data
assumptions
Effects of
interventions
Causes of
effects
Explanations
Causal analysis deals with changes (dynamics)
Experiments

8. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT)
Spurious correlation
Randomization
Confounding / Effect
Instrument
Holding constant
Explanatory variables
STATISTICAL
Regression
Association / Independence
“Controlling for” / Conditioning
Odd and risk ratios
Collapsibility
Causal and statistical concepts do not mix.

9. } FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) 
Spurious correlation
Randomization
Confounding / Effect
Instrument
Holding constant
Explanatory variables
STATISTICAL
Regression
Association / Independence
“Controlling for” / Conditioning
Odd and risk ratios
Collapsibility
Causal and statistical concepts do not mix.
No causes in – no causes out (Cartwright, 1989)
statistical assumptions + data
causal assumptions
causal conclusions  }
Causal assumptions cannot be expressed in the mathematical language of standard statistics.

10. } FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) 
Spurious correlation
Randomization
Confounding / Effect
Instrument
Holding constant
Explanatory variables
STATISTICAL
Regression
Association / Independence
“Controlling for” / Conditioning
Odd and risk ratios
Collapsibility
Causal and statistical concepts do not mix.
No causes in – no causes out (Cartwright, 1989)
statistical assumptions + data
causal assumptions
causal conclusions  }
Causal assumptions cannot be expressed in the mathematical language of standard statistics.
Non-standard mathematics:
Structural equation models (SEM)
Counterfactuals (Neyman-Rubin)
Causal Diagrams (Wright, 1920)

11. WHAT'S IN A CAUSAL MODEL? Oracle that assigns truth value to causal
sentences:
Action sentences: B if we do A.
Counterfactuals: B would be different if
A were true.
Explanation: B occurred because of A.
Optional: with what probability?

12. ORACLE FOR MANIPILATION
FAMILIAR CAUSAL MODEL
ORACLE FOR MANIPILATION X Y
Here is a causal model we all remember from high-school -- a circuit diagram.
There are 4 interesting points to notice in this example:
(1) It qualifies as a causal model -- because it contains the information to confirm or refute all action, counterfactual and explanatory sentences concerned with the operation of the circuit.
For example, anyone can figure out what the output would be like if we set Y to zero, or if we change this OR gate to a NOR gate or if we perform any of the billions combinations of such actions.
(2) Logical functions (Boolean input-output relation) is insufficient for answering such queries
(3)These actions were not specified in advance, they do not have special names and they do not show up in the diagram.
In fact, the great majority of the action queries that this circuit can answer have never been considered by the designer of this circuit.
(4) So how does the circuit encode this extra information?
Through two encoding tricks:
4.1 The symbolic units correspond to stable physical mechanisms
(i.e., the logical gates)
4.2 Each variable has precisely one mechanism that determines its value. Z
INPUT
OUTPUT

13. CAUSAL MODELS AND CAUSAL DIAGRAMS
Definition: A causal model is a 3-tuple
M = V,U,F
with a mutilation operator do(x): M Mx where:
(i) V = {V1…,Vn} endogenous variables,
(ii) U = {U1,…,Um} background variables
(iii) F = set of n functions, fi : V \ Vi  U  Vi
vi = fi(pai,ui) PAi  V \ Vi Ui  U

14. CAUSAL MODELS AND CAUSAL DIAGRAMS U1 I W U2 Q P PAQ
Definition: A causal model is a 3-tuple
M = V,U,F
with a mutilation operator do(x): M Mx where:
(i) V = {V1…,Vn} endogenous variables,
(ii) U = {U1,…,Um} background variables
(iii) F = set of n functions, fi : V \ Vi  U  Vi
vi = fi(pai,ui) PAi  V \ Vi Ui  U U1 I W U2 Q P
PAQ

15. CAUSAL MODELS AND MUTILATION Definition: A causal model is a 3-tuple
M = V,U,F
with a mutilation operator do(x): M Mx where:
(i) V = {V1…,Vn} endogenous variables,
(ii) U = {U1,…,Um} background variables
(iii) F = set of n functions, fi : V \ Vi  U  Vi
vi = fi(pai,ui) PAi  V \ Vi Ui  U
(iv) Mx= U,V,Fx, X  V, x  X
where Fx = {fi: Vi  X }  {X = x}
(Replace all functions fi corresponding to X with the constant functions X=x)

16. CAUSAL MODELS AND MUTILATION U1 I W U2 Q P
Definition: A causal model is a 3-tuple
M = V,U,F
with a mutilation operator do(x): M Mx where:
(i) V = {V1…,Vn} endogenous variables,
(ii) U = {U1,…,Um} background variables
(iii) F = set of n functions, fi : V \ Vi  U  Vi
vi = fi(pai,ui) PAi  V \ Vi Ui  U
(iv) U1 I W U2 Q P

17. CAUSAL MODELS AND MUTILATION U1 I W U2 Q P P = p0
Definition: A causal model is a 3-tuple
M = V,U,F
with a mutilation operator do(x): M Mx where:
(i) V = {V1…,Vn} endogenous variables,
(ii) U = {U1,…,Um} background variables
(iii) F = set of n functions, fi : V \ Vi  U  Vi
vi = fi(pai,ui) PAi  V \ Vi Ui  U
(iv) Mp U1 I W U2 Q P
P = p0

18. PROBABILISTIC CAUSAL MODELS Definition: A causal model is a 3-tuple
M = V,U,F
with a mutilation operator do(x): M Mx where:
(i) V = {V1…,Vn} endogenous variables,
(ii) U = {U1,…,Um} background variables
(iii) F = set of n functions, fi : V \ Vi  U  Vi
vi = fi(pai,ui) PAi  V \ Vi Ui  U
(iv) Mx= U,V,Fx, X  V, x  X
where Fx = {fi: Vi  X }  {X = x}
(Replace all functions fi corresponding to X with the constant functions X=x)
Definition (Probabilistic Causal Model):
M, P(u)
P(u) is a probability assignment to the variables in U.

19. CAUSAL MODELS AND COUNTERFACTUALS
Definition: Potential Response
The sentence: “Y would be y (in unit u), had X been x,”
denoted Yx(u) = y, is the solution for Y in a mutilated model
Mx, with the equations for X replaced by X = x.
(“unit-based potential outcome”)

20. CAUSAL MODELS AND COUNTERFACTUALS
Definition: Potential Response
The sentence: “Y would be y (in unit u), had X been x,”
denoted Yx(u) = y, is the solution for Y in a mutilated model
Mx, with the equations for X replaced by X = x.
(“unit-based potential outcome”)
Joint probabilities of counterfactuals:

21. CAUSAL MODELS AND COUNTERFACTUALS
Definition: Potential Response
The sentence: “Y would be y (in unit u), had X been x,”
denoted Yx(u) = y, is the solution for Y in a mutilated model
Mx, with the equations for X replaced by X = x.
(“unit-based potential outcome”)
Joint probabilities of counterfactuals:
In particular:

22. 3-STEPS TO COMPUTING COUNTERFACTUALS Abduction TRUE TRUE U
S5. If the prisoner is dead, he would still be dead
if A were not to have shot. DDA
Abduction
TRUE
TRUE U
(Court order) C
(Captain)
Consider now our counterfactual sentence
S5: If the prisoner is Dead, he would still be dead if A were not to have shot. D ==> DA
The antecedant {A} should still be treated as interventional surgery, but only after we fully account for the evidence given: D.
This calls for three steps
1 Abduction: Interpret the past in light of the evidence
2. Action: Bend the course of history (minimally) to account for the hypothetical antecedant (A).
3.Prediction: Project the consequences to the future. A B
(Riflemen) D
(Prisoner)

23. 3-STEPS TO COMPUTING COUNTERFACTUALS Abduction Action Prediction TRUE
S5. If the prisoner is dead, he would still be dead
if A were not to have shot. DDA
Abduction
TRUE U D B C A
FALSE
TRUE
Action
TRUE U D B C A
FALSE
Prediction
TRUE U C A B D

24. COMPUTING PROBABILITIES
OF COUNTERFACTUALS
P(S5). The prisoner is dead. How likely is it that he would be dead
if A were not to have shot. P(DA|D) = ?
Abduction
TRUE
Action U D B C A
FALSE
P(u|D)
Prediction U D B C A
FALSE
P(u|D)
P(DA|D)
P(u) U
P(u|D) C
Suppose we are not entirely ignorant of U, but can assess the degree of belief P(u).
The same 3-steps apply to the computation of the counterfactual probability (that the prisoner be dead if A were not to have shot)
The only difference is that we now use the evidence to update P(u) into P(u|e), and draw probabilistic instead of logical conclusions. A B D

25. CAUSAL INFERENCE MADE EASY (1985-2000)
Inference with Nonparametric Structural Equations
made possible through Graphical Analysis.
Mathematical underpinning of counterfactuals
through nonparametric structural equations
Graphical-Counterfactuals symbiosis

26. IDENTIFIABILITY P(M1) = P(M2) Þ Q(M1) = Q(M2) Definition:
Let Q(M) be any quantity defined on a causal
model M, and let A be a set of assumption.
Q is identifiable relative to A iff
P(M1) = P(M2) Þ Q(M1) = Q(M2)
for all M1, M2, that satisfy A.

27. IDENTIFIABILITY P(M1) = P(M2) Þ Q(M1) = Q(M2) Definition:
Let Q(M) be any quantity defined on a causal
model M, and let A be a set of assumption.
Q is identifiable relative to A iff
P(M1) = P(M2) Þ Q(M1) = Q(M2)
for all M1, M2, that satisfy A.
In other words, Q can be determined uniquely
from the probability distribution P(v) of the
endogenous variables, V, and assumptions A.

28. IDENTIFIABILITY P(M1) = P(M2) Þ Q(M1) = Q(M2) Definition:
Let Q(M) be any quantity defined on a causal
model M, and let A be a set of assumption.
Q is identifiable relative to A iff
P(M1) = P(M2) Þ Q(M1) = Q(M2)
for all M1, M2, that satisfy A.
In this talk:
A: Assumptions encoded in the diagram
Q1: P(y|do(x)) Causal Effect (= P(Yx=y))
Q2: P(Yx =y | x, y) Probability of necessity
Q3: Direct Effect

29. THE FUNDAMENTAL THEOREM
OF CAUSAL INFERENCE
Causal Markov Theorem:
Any distribution generated by Markovian structural model M
(recursive, with independent disturbances) can be factorized as
Where pai are the (values of) the parents of Vi in the causal
diagram associated with M.

30. THE FUNDAMENTAL THEOREM
OF CAUSAL INFERENCE
Causal Markov Theorem:
Any distribution generated by Markovian structural model M
(recursive, with independent disturbances) can be factorized as
Where pai are the (values of) the parents of Vi in the causal
diagram associated with M.
Corollary: (Truncated factorization, Manipulation Theorem)
The distribution generated by an intervention do(X=x)
(in a Markovian model M) is given by the truncated factorization

31. RAMIFICATIONS OF THE FUNDAMENTAL THEOREM
U (unobserved)
X = x Z Y
Smoking
Tar in
Lungs
Cancer X
Given P(x,y,z), should we ban smoking?
Pre-intervention
Post-intervention

32. RAMIFICATIONS OF THE FUNDAMENTAL THEOREM
U (unobserved)
X = x Z Y
Smoking
Tar in
Lungs
Cancer X
Given P(x,y,z), should we ban smoking?
Pre-intervention
Post-intervention

33. RAMIFICATIONS OF THE FUNDAMENTAL THEOREM
U (unobserved)
X = x Z Y
Smoking
Tar in
Lungs
Cancer X
Given P(x,y,z), should we ban smoking?
Pre-intervention
Post-intervention
To compute P(y,z|do(x)), we must eliminate u. (graphical problem).

34. THE BACK-DOOR CRITERION
Graphical test of identification
P(y | do(x)) is identifiable in G if there is a set Z of
variables such that Z d-separates X from Y in Gx. G Gx Z1 Z1 Z2 Z2 Z Z3 Z3 Z4 Z5 Z5 Z4 X Z6 Y X Z6 Y

35. THE BACK-DOOR CRITERION
Graphical test of identification
P(y | do(x)) is identifiable in G if there is a set Z of
variables such that Z d-separates X from Y in Gx. G Gx Z1 Z1 Z2 Z2 Z Z3 Z3 Z4 Z5 Z5 Z4 X Z6 Y X Z6 Y
Moreover, P(y | do(x)) = å P(y | x,z) P(z)
(“adjusting” for Z) z

36. RULES OF CAUSAL CALCULUS
Rule 1: Ignoring observations
P(y | do{x}, z, w) = P(y | do{x}, w)
Rule 2: Action/observation exchange
P(y | do{x}, do{z}, w) = P(y | do{x},z,w)
Rule 3: Ignoring actions
P(y | do{x}, do{z}, w) = P(y | do{x}, w)

37. DERIVATION IN CAUSAL CALCULUS
Genotype (Unobserved)
Smoking
Tar
Cancer
P (c | do{s}) = t P (c | do{s}, t) P (t | do{s})
Probability Axioms
= t P (c | do{s}, do{t}) P (t | do{s})
Rule 2
= t P (c | do{s}, do{t}) P (t | s)
Rule 2
= t P (c | do{t}) P (t | s)
Rule 3
= st P (c | do{t}, s) P (s | do{t}) P(t |s)
Probability Axioms
Rule 2
= st P (c | t, s) P (s | do{t}) P(t |s)
= s t P (c | t, s) P (s) P(t |s)
Rule 3

38. OUTLINE Modeling: Statistical vs. Causal
Causal models and identifiability
Inference to three types of claims:
Effects of potential interventions,
Claims about attribution (responsibility)

39. DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem)
Your Honor! My client (Mr. A) died BECAUSE
he used that drug.

40. DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem)
Your Honor! My client (Mr. A) died BECAUSE
he used that drug.
Court to decide if it is MORE PROBABLE THAN
NOT that A would be alive BUT FOR the drug!
P(? | A is dead, took the drug) > 0.50

41. THE PROBLEM Theoretical Problems: What is the meaning of PN(x,y):
“Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”

42. THE PROBLEM Theoretical Problems: What is the meaning of PN(x,y):
“Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”
Answer:

43. THE PROBLEM Theoretical Problems: What is the meaning of PN(x,y):
“Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”
Under what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined.

44. WHAT IS INFERABLE FROM EXPERIMENTS?
Simple Experiment:
Q = P(Yx= y | z)
Z nondescendants of X.
Compound Experiment:
Q = P(YX(z) = y | z)
Multi-Stage Experiment:
etc…

45. CAN FREQUENCY DATA DECIDE LEGAL RESPONSIBILITY?
Experimental Nonexperimental
do(x) do(x) x x
Deaths (y)
Survivals (y)
1,000 1,000 1,000 1,000
Nonexperimental data: drug usage predicts longer life
Experimental data: drug has negligible effect on survival
Plaintiff: Mr. A is special.
He actually died
He used the drug by choice
Court to decide (given both data):
Is it more probable than not that A would be alive
but for the drug?

46. TYPICAL THEOREMS (Tian and Pearl, 2000)
Bounds given combined nonexperimental and experimental data
Identifiability under monotonicity (Combined data)
corrected Excess-Risk-Ratio

47. SOLUTION TO THE ATTRIBUTION PROBLEM (Cont)
WITH PROBABILITY ONE P(yx | x,y) =1
From population data to individual case
Combined data tell more that each study alone

48. OUTLINE Modeling: Statistical vs. Causal
Causal models and identifiability
Inference to three types of claims:
Effects of potential interventions,
Claims about attribution (responsibility)
Claims about direct and indirect effects

49. QUESTIONS ADDRESSED
What is the semantics of direct and indirect effects?
Can we estimate them from data? Experimental data?

50. WHY DECOMPOSE
EFFECTS?
Direct (or indirect) effect may be more transportable.
Indirect effects may be prevented or controlled.
Direct (or indirect) effect may be forbidden
Pill 
Pregnancy + +
Thrombosis
Gender
Qualification
Hiring

51. TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE SEMANTICS
IN LINEAR MODELS b X Z
z = bx + 1
y = ax + cz + 2 a c Y
a + bc a bc

52. SEMANTICS BECOMES NONTRIVIAL IN NONLINEAR MODELS
(even when the model is completely specified) X Z
z = f (x, 1)
y = g (x, z, 2) Y
Dependent on z?
Void of operational meaning?

53. THE OPERATIONAL MEANING OF
DIRECT EFFECTS X Z
z = f (x, 1)
y = g (x, z, 2) Y
“Natural” Direct Effect of X on Y:
The expected change in Y per unit change of X, when we keep Z constant at whatever value it attains before the change.
In linear models, NDE = Controlled Direct Effect

54. POLICY IMPLICATIONS (Who cares?) indirect
What is the direct effect of X on Y?
The effect of Gender on Hiring if sex discrimination
is eliminated.
GENDER
QUALIFICATION
HIRING X Z
IGNORE f Y

55. THE OPERATIONAL MEANING OF
INDIRECT EFFECTS X Z
z = f (x, 1)
y = g (x, z, 2) Y
“Natural” Indirect Effect of X on Y:
The expected change in Y when we keep X constant, say at x0, and let Z change to whatever value it would have under a unit change in X.
In linear models, NIE = TE - DE

56. LEGAL DEFINITIONS TAKE THE NATURAL CONCEPTION
(FORMALIZING DISCRIMINATION)
``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’
[Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))]
x = male, x = female
y = hire, y = not hire
z = applicant’s qualifications
NO DIRECT EFFECT

57. SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS
Consider the quantity
Given M, P(u), Q is well defined
Given u, Zx*(u) is the solution for Z in Mx*, call it z
is the solution for Y in Mxz
Can Q be estimated from data?

58. GRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION
OF AVERAGE NATURAL DIRECT EFFECTS
Theorem: If there exists a set W such that
Example:

59. HOW THE PROOF GOES? Proof:
Each factor is identifiable by experimentation.

60. GRAPHICAL CRITERION FOR COUNTERFACTUAL INDEPENDENCEX Z X Z U1 Y U1 Y U3 U1 X Y U2 Z

61. GRAPHICAL CONDITION FOR NONEXPERIMENTAL IDENTIFICATION
OF AVERAGE NATURAL DIRECT EFFECTS
Identification conditions
There exists a W such that (Y Z | W)GXZ
There exist additional covariates that render all
counterfactual terms identifiable.

62. IDENTIFICATION IN MARKOVIAN MODELS Corollary 3:
The average natural direct effect in Markovian models is
identifiable from nonexperimental data, and it is given by X Z Y

63. RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT EFFECTS
Theorem 5: The total, direct and indirect effects obey
The following equality
In words, the total effect (on Y) associated with the
transition from x* to x is equal to the difference
between the direct effect associated with this transition
and the indirect effect associated with the reverse
transition, from x to x*.

64. GENERAL PATH-SPECIFIC
EFFECTS (Def.) X x* X W Z W Z
z* = Zx* (u) Y Y
Form a new model, , specific to active subgraph g
Definition: g-specific effect
Nonidentifiable even in Markovian models

65. ANSWERS TO QUESTIONS Graphical conditions for estimability from
experimental / nonexperimental data.
Graphical conditions hold in Markovian models

66. ANSWERS TO QUESTIONS Graphical conditions for estimability from
experimental / nonexperimental data.
Graphical conditions hold in Markovian models
Useful in answering new type of policy questions
involving mechanism blocking instead of variable fixing.

67. THE OVERRIDING THEME Define Q(M) as a counterfactual expression
Determine conditions for the reduction
If reduction is feasible, Q is inferable.
Demonstrated on three types of queries:
Q1: P(y|do(x)) Causal Effect (= P(Yx=y))
Q2: P(Yx = y | x, y) Probability of necessity
Q3: Direct Effect

68. THE COUNTERFACTUAL TEST
ACTUAL CAUSATION AND
THE COUNTERFACTUAL TEST
"We may define a cause to be an object followed by
another,..., where, if the first object had not been, the
second never had existed."
Hume, Enquiry, 1748
Lewis (1973): "x CAUSED y " if x and y are true, and
y is false in the closest non-x-world.
Structural interpretation:
(i) X(u)=x
(ii) Y(u)=y
(iii) Yx (u)  y for x   x
The modern formulation of this concept start again with David Hume. It was given a possible-world semantics by David Lewis, and even simpler semantics using our structural-interpretation of counterfactuals.
Notice how we write, in surgery language, the sentence:
"If the first object (x) had not been, the second (y) never had existed."
Yx'(u)  y for x'  x
Meaning: The solution for Y in a model mutilated by the operator do(X=x') is not equal to y.
But this definition of "cause" is known to be ridden with problems.

69. COUNTERFACTUAL DEFINITION
PROBLEM WITH THE
COUNTERFACTUAL DEFINITION
Back-up to shoot iff Captain does not shoot at 12:00 noon
(Captain) X W
(Back-up)
(Prisoner) Y

70. COUNTERFACTUAL DEFINITION
PROBLEM WITH THE
COUNTERFACTUAL DEFINITION
Back-up to shoot iff Captain does not shoot at 12:00 noon
Scenario: Captain shot before noon
Prisoner is dead
= 1
= 0
(Captain) X W
(Back-up)
(Prisoner) Y

71. COUNTERFACTUAL DEFINITION
PROBLEM WITH THE
COUNTERFACTUAL DEFINITION
Back-up to shoot iff Captain does not shoot at 12:00 noon
Scenario: Captain shot before noon
Prisoner is dead
Is Captain’s shot the cause of death?
Yes, but the counterfactual test fails!
Intuition: Back-up might fall asleep – structural contingency
= 1
= 0
(Captain) X W
(Back-up)
(Prisoner) Y

72. SELECTED STRUCTURAL CONTINGENCIES AND SUSTENANCE
x sustains y against W iff:
(i) X(u) = x;
(ii) Y(u) = y ;
(iii) Yxw(u) = y for all w; and
(iv) Yx w (u) = y for some x  x and some w X
= 1 X
= 0 W
= w = 0 W
= w = 0 Y
= 1 Y
= 0

73. EXPLANATORY POWER (Halpern and Pearl, 2001)
Definition: The explanatory power of a proposition
X=x relative to an observed event Y=y is given by
P(K{x,y}|x), the pre-discovery probability of the set
of contexts K in which x is the actual cause of y.
FIRE
AND
Oxygen
Match
Oxygen
KO,F = KM,F = K
Match
EP(O) = P(K|O) << 1 EP(M) = P(K|M) » 1

74. CORRECTNESS and CORROBORATIONP*
P*(S)
Falsifiability: P*(S)  P*
D (Data)
Constraints implied by S
Data D corroborates structure S if S is (i) falsifiable
and (ii) compatible with D.
Types of constraints: 1. conditional independencies 2. inequalities (for restricted domains) 3. functional
e.g., w x y z

75. FROM CORROBORATING MODELS TO CORROBORATING CLAIMS
e.g., An un-corroborated structure, a is identifiable. x x y y
a = rYX a
Intuitively, claim a = rYX is not corroborated because
the assumptions that entail the claim are not falsifiable. i.e., no data falsifies the assumption rs = 0. x y a rs
a = rYX - rs

76. FROM CORROBORATING MODELS TO CORROBORATING CLAIMS
A corroborated structure can imply uncorroborated
claims.
e.g., x x y y z a x y z a

77. FROM CORROBORATING MODELS TO CORROBORATING CLAIMS
Some claims can be more corroborated than others.
e.g., x x y y z a x y z a b
b = rZY is corroborated because the assumptions
needed for entailing this claim constrain the data by a

78. FROM CORROBORATING MODELS TO CORROBORATING CLAIMS
Some claims can be more corroborated than others.
e.g., x x y y z x y z a a b a a b
Definition:
An identifiable claim C is corroborated by data if the
union of all minimal sets of assumptions sufficient
for identifying C is corroborated by the data.

79. GRAPHICAL CRITERION FOR
CORROBORATED CLAIMS
Theorem:
An identifiable claim C is corroborated by data if
The intersection of all maximal supergraphs sufficient
for identifying C is corroborated by the data.
e.g., x x x y y z x y z a a b a b

80. GRAPHICAL CRITERION FOR
CORROBORATED CLAIMS
Theorem:
An identifiable claim C is corroborated by data if
The intersection of all maximal supergraphs sufficient
for identifying C is corroborated by the data.
e.g., b a x y z
Maximal supergraphs: x x x y y z x x x y y z x y z a b a b a a b
Intersection: x x x y y z a b

81. CONCLUSIONS Structural-model semantics enriched
with logic + graphs leads to formal
interpretation and practical assessments
of wide variety of causal and counterfactual
relationships.