1. Department of Computer Science
THE MATHEMATICS
OF CAUSAL INFERENCE
IN STATISTICS
Judea Pearl
Department of Computer Science
UCLA

2. OUTLINE Statistical vs. Causal Modeling:
distinction and mental barriers
N-R vs. structural model:
strengths and weaknesses
Formal semantics for counterfactuals:
definition, axioms, graphical representations
Graphs and Algebra: Symbiosis
translation and accomplishments

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

4. FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
Probability and statistics deal with static relations P
Joint
Distribution P
Joint
Distribution
Q(P)
(Aspects of P)
Data
change
Inference
What happens when P changes?
e.g.,
Infer whether customers who bought product A
would still buy A if we were to double the price.

5. FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
What remains invariant when P changes say, to satisfy
P (price=2)=1 P
Joint
Distribution P
Joint
Distribution
Q(P)
(Aspects of P)
Data
change
Inference
Note: P (v)  P (v | price = 2)
P does not tell us how it ought to change
e.g. Curing symptoms vs. curing diseases
e.g. Analogy: mechanical deformation

6. 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.

7. } 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.

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.
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 (Wright, 1920; Simon, 1960)
Counterfactuals (Neyman-Rubin (Yx), Lewis (x Y))

9. TWO PARADIGMS FOR CAUSAL INFERENCE Observed: P(X, Y, Z,...)
Conclusions needed: P(Yx=y), P(Xy=x | Z=z)...
How do we connect observables, X,Y,Z,…
to counterfactuals Yx, Xz, Zy,… ?
N-R model
Counterfactuals are
primitives, new variables
Super-distribution
P*(X, Y,…, Yx, Xz,…)
X, Y, Z constrain Yx, Zy,…
Structural model
Counterfactuals are
derived quantities
Subscripts modify the
distribution
P(Yx=y) = Px(Y=y)

10. “SUPER” DISTRIBUTION IN N-R MODEL X Y Z Yx=0 Yx=1 Xz=0 Xz=1 Xy=0 U1 Y 1 Z 1
Yx=0 1
Yx=1 1
Xz=0 1
Xz=1 1
Xy=0
0
1 U u1 u2 u3 u4
inconsistency: x = 0  Yx=0 = Y
Y = xY1 + (1-x) Y0

11. TYPICAL INFERENCE IN N-R MODEL Find P*(Yx=y) given covariate Z,
Assume ignorability:
Yx  X | Z 
Assume consistency:
X=x  Yx=Y
Problems: 
Yx  X | Z judgmental &
opaque
Try it: X  Y  Z ? 1) 
Yx  X | Z judgmental &
opaque
X, Y and Yx? 2)
Is consistency the only connection between
Is consistency the only connection between

12. THE STRUCTURAL MODEL PARADIGM Data Joint Generating Q(M) Distribution
(Aspects of M)
Data
Inference
M – Oracle for computing answers to Q’s.
e.g.,
Infer whether customers who bought product A
would still buy A if we were to double the price.

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. STRUCTURAL CAUSAL MODELS
Definition: A structural causal model is a 4-tuple
V,U, F, P(u), where
V = {V1,...,Vn} are observable variables
U = {U1,...,Um} are background variables
F = {f1,..., fn} are functions determining V,
vi = fi(v, u)
P(u) is a distribution over U
P(u) and F induce a distribution P(v) over observable variables

15. STRUCTURAL MODELS AND CAUSAL DIAGRAMS U1 I W U2 Q P PAQ
The arguments of the functions vi = fi(v,u) define a graph
vi = fi(pai,ui) PAi  V \ Vi Ui  U
Example: Price – Quantity equations in economics U1 I W U2 Q P
PAQ

16. STRUCTURAL MODELS AND INTERVENTION U1 I W U2 Q P
Let X be a set of variables in V.
The action do(x) sets X to constants x regardless of
the factors which previously determined X.
do(x) replaces all functions fi determining X with the
constant functions X=x, to create a mutilated model Mx U1 I W U2 Q P

17. STRUCTURAL MODELS AND INTERVENTION U1 I W U2 Q P P = p0
Let X be a set of variables in V.
The action do(x) sets X to constants x regardless of
the factors which previously determined X.
do(x) replaces all functions fi determining X with the
constant functions X=x, to create a mutilated model Mx Mp U1 I W U2 Q P
P = p0

18. CAUSAL MODELS AND COUNTERFACTUALS
Definition:
The sentence: “Y would be y (in situation u),
had X been x,” denoted Yx(u) = y, means:
The solution for Y in a mutilated model Mx,
(i.e., the equations for X replaced by X = x)
with input U=u, is equal to y.
Joint probabilities of counterfactuals:
The super-distribution P* is derived from M.
Parsimonous, consistent, and transparent

19. AXIOMS OF CAUSAL COUNTERFACTUALS
Y would be y, had X been x (in state U = u)
Definiteness
Uniqueness
Effectiveness
Composition
Reversibility

20. GRAPHICAL – COUNTERFACTUALS SYMBIOSIS
Every causal graph expresses counterfactuals
assumptions, e.g., X  Y  Z
Missing arrows Y  Z
2. Missing arcs Y Z
consistent, and readable from the graph.
Every theorem in SEM is a theorem in N-R,
and conversely.

21. STRUCTURAL ANALYSIS: SOME USEFUL RESULTS
Complete formal semantics of counterfactuals
Transparent language for expressing assumptions
Complete solution to causal-effect identification
Legal responsibility (bounds)
Non-compliance (universal bounds)
Integration of data from diverse sources
Direct and Indirect effects,
Complete criterion for counterfactual testability

22. REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY)
Regression (claimless):
Y = ax + a1z1 + a2z akzk + y
a E[Y | x,z] / x = Ryx z Y y | X,Z
Structural (empirical, falsifiable):
Y = bx + b1z1 + b2z bkzk + uy
b E[Y | do(x,z)] / x = E[Yx,z] / x
Assumptions: Yx,z = Yx,z,w
cov(ui, uj) = 0 for some i,j =   = 

23. REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY)
Regression (claimless):
Y = ax + a1z1 + a2z akzk + y
a E[Y | x,z] / x = Ryx z Y y | X,Z
Structural (empirical, falsifiable):
Y = bx + b1z1 + b2z bkzk + uy
b E[Y | do(x,z)] / x = E[Yx,z] / x
Assumptions: Yx,z = Yx,z,w
cov(ui, uj) = 0 for some i,j  =  = 
The mother of all questions:
“When would b equal a?,” or
“What kind of regressors should Z include?”
Answer: When Z satisfies the backdoor criterion
Question: When is b estimable by regression methods?
Answer: graphical criteria available

24. 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

25. RECENT RESULTS ON IDENTIFICATION
do-calculus is complete
Complete graphical criterion for identifying
causal effects (Shpitser and Pearl, 2006).
Complete graphical criterion for empirical
testability of counterfactuals
(Shpitser and Pearl, 2007).

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

27. 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
PN =

28. THE PROBLEM Semantical Problem: 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.”

29. THE PROBLEM Semantical Problem: 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:
Computable from M

30. THE PROBLEM Semantical Problem: 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.
Analytical Problem:

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

32. 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?

33. SOLUTION TO THE ATTRIBUTION PROBLEM
WITH PROBABILITY ONE 1  P(yx | x,y)  1
Combined data tell more that each study alone

34. CONCLUSIONS Structural-model semantics, enriched with logic
and graphs, provides:
Complete formal basis for the N-R model
Unifies the graphical, potential-outcome and structural equation approaches
Powerful and friendly causal calculus
(best features of each approach)