1. Computer Science and Statistics
CAUSAL MODELING
AND THE
LOGIC OF SCIENCE
Judea Pearl
Computer Science and Statistics
UCLA

2. Scope and Language in Scientific Theories
OVERVIEW
Scope and Language in Scientific Theories
Statistical models
(observtions, PL)
Causal models
2.1 Stochastic causal model
(interventions, PL + modality)
2.2 Functional causal models
(counterfactuals, PL + subjunctives)
General equational models
(explicit interventions, PL)
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
General Scientific theories
(objects-properties, FOL-SOL ...)

3. 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
Falsifiability and Corroboration

4. 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)

5. 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.

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

7. 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

8. 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

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.

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.

11. } 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)

12. 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?

13. 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

14. 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

15. 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

16. 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)

17. 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

18. 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

19. 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.

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”)

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:

22. 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:

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
Action
Prediction
TRUE U U D B C A
FALSE
TRUE
TRUE U D B C A
FALSE C
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 D
TRUE

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
Action
Prediction
P(u) U U D B C A
FALSE
P(u|D) U D B C A
FALSE
P(u|D)
P(DA|D)
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
TRUE

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?

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. 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

51. 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?

52. 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

53. 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

54. 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

55. 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
YxZx = Yx, YxZx = Yx

56. 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?

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

58. 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.

59. 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

60. FALSIFIABILITY and CORROBORATIONP*
P*(M)
Falsifiability: P*(M)  P*
D (Data)
Constraints implied by M
Data D corroborates model M if M 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

61. Changes under interventions
OTHER TESTABLE CLAIMS
Changes under interventions
For all causal models:
For all semi-Markovian models:
For Markovian models (and ):
For a given Markovian model:

62. FROM CORROBORATING MODELS TO CORROBORATING CLAIMS
A corroborated model can imply identifiable yet
uncorroborated claims.
e.g., x x y y z z x y z a a b
Some claims can be more corroborated than others.
Definition:
An identifiable claim C is corroborated by data if some minimal set of assumptions in M sufficient for identifying C is corroborated by the data.
Graphical criterion: minimal submodel = maximal supergraph

63. FROM CORROBORATING MODELS TO CORROBORATING CLAIMS
A corroborated model can imply identifiable yet
uncorroborated claims.
e.g., x x y y z z x y z a a b
Some claims can be more corroborated than others.
Definition:
An identifiable claim C is corroborated by data if some minimal set of assumptions in M sufficient for identifying C is corroborated by the data.
Graphical criterion: minimal submodel = maximal supergraph

64. Scope and Language in Scientific Theories
OVERVIEW
Scope and Language in Scientific Theories
Statistical models
(observtions, PL)
Causal models
2.1 Stochastic causal model
(interventions, PL + modality)
2.2 Functional causal models
(counterfactuals, PL + subjunctives)
General equational models
(explicit interventions, PL)
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
General Scientific theories
(objects-properties, FOL-SOL ...)