1. A MACHINE LEARNING EXERCISE
CAUSAL INFERENCE AS
A MACHINE LEARNING EXERCISE
Judea Pearl
Computer Science and Statistics
UCLA

2. OUTLINE Learning: Statistical vs. Causal concepts
Causal models and identifiability
Learnability of three types of causal queries:
Effects of potential interventions,
Queries about attribution (responsibility)
Queries about direct and indirect effects

3. TRADITIONAL MACHINE LEARNING PARADIGM P Joint Distribution Q(P)
(Aspects of P)
Data
Learning
e.g.,
Learn whether customers who bought product A
would also buy product B.
Q = P(B|A)

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

5. THE SECRETS OF CAUSAL MODELS
Causal Model = Data-generating model satisfying:
Modularity (Symbol-mechanism correspondence)
Uniqueness (Variable-mechanism correspondence)

6. THE SECRETS OF CAUSAL MODELS
Causal Model = Data-generating model satisfying:
Modularity (Symbol-mechanism correspondence)
Uniqueness (Variable-mechanism correspondence)
Causal Model Joint Distribution
PR = f (CP,PS, e1) P(PR, PS, CP, ME, QS)
QS = g(ME, PR, e2); P(e1, e2)
Q1: P(QS|PR=2) computable from P (and M)
Q2: P(QS|do(PR=2) computable from M (not P) f
Quantity
Sold (QS) g
Cost Proj.
Prev. Sale
Others (e1)
Others (e2)
Marketing (MS)
PRICE (PR)

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

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 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

9. FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
Data
joint
distribution
predictions
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

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
Propensity score
Causal and statistical concepts do not mix.

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
Propensity score
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 Z
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.
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 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. 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

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. 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

24. NON-PARAMETRIC STRUCTURAL MODELS
Given P(x,y,z), should we ban smoking? U U 1 1 U U 3 3 U U 2 2 f3 f1 f2   X Z Y X Z Y
Smoking
Tar in
Lungs
Cancer
Smoking
Tar in
Lungs
Cancer
Linear Analysis Nonparametric Analysis
x = u1,
z = x + u2,
y = z +  u1 + u3.
x = f1(u1),
z = f2(x, u2),
y = f3(z, u1, u3).
Find:    Find: P(y|do(x))

25. LEARNING THE EFFECTS OF ACTIONS Given P(x,y,z), should we ban smoking?1 1 U U 3 3 U U 2 2 f3 f2  
X = x Y X Z Z Y
Smoking
Tar in
Lungs
Cancer
Smoking
Tar in
Lungs
Cancer
Linear Analysis Nonparametric Analysis
x = u1,
z = x + u2,
y = z +  u1 + u3.
x = const.
z = f2(x, u2),
y = f3(z, u1, u3). 
Find:    Find: P(y|do(x)) = P(Y=y) in new model

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) if ( Y ^ ^ Z | X , W ) G X Z

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. IDENTIFICATION RESULT
A RECENT
IDENTIFICATION RESULT
Theorem: [Tian and Pearl, 2001]
The causal effect P(y|do(x)) is identifiable whenever
the ancestral graph of Y contains no confounding
path ( ) between X and any of
its children. X Y X Z1
(c) Z2 X Z1 Z1 Z2 Y Y
(a)
(b)

39. OUTLINE Learning: Statistical vs. Causal concepts
Causal models and identifiability
Learnability of three types of causal queries:
Distinguishing direct from indirect effects
Queries about attribution (responsibility)

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

41. 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

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.”

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.”
Answer:

44. 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.

45. WHAT IS LEARNABLE 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…

46. 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?

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

48. 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

49. OUTLINE Learning: Statistical vs. Causal concepts
Causal models and identifiability
Learnability of three types of causal queries:
Effects of potential interventions,
Queries about attribution (responsibility)
Queries about direct and indirect effects

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

51. 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

52. 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

53. 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

54. 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?

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

56. IDENTIFICATION IN
MARKOVIAN MODELS X Z Y

57. 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.

58. CONCLUSIONS General theme: Define Q(M) as a counterfactual expression
Determine conditions for the reduction
If reduction is feasible, Q is learnable.
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