Transcript CAUSAL INFERENCE AS A MACHINE LEARNING EXERCISE

THE MATHEMATICS
OF CAUSAL INFERENCE
IN STATISTICS
Judea Pearl
Department of Computer Science
UCLA
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
TRADITIONAL STATISTICAL
INFERENCE PARADIGM
Data
P
Joint
Distribution
Q(P)
(Aspects of P)
Inference
e.g.,
Infer whether customers who bought product A
would also buy product B.
Q = P(B | A)
FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
Probability and statistics deal with static relations
Data
P
Joint
Distribution
P
Joint
Distribution
change
Q(P)
(Aspects of P)
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.
FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES
What remains invariant when P changes say, to satisfy
P (price=2)=1
Data
P
Joint
Distribution
P
Joint
Distribution
change
Q(P)
(Aspects of P)
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
FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES (CONT)
1. Causal and statistical concepts do not mix.
CAUSAL
Spurious correlation
Randomization
Confounding / Effect
Instrument
Holding constant
Explanatory variables
2.
3.
4.
STATISTICAL
Regression
Association / Independence
“Controlling for” / Conditioning
Odd and risk ratios
Collapsibility
FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES (CONT)
1. Causal and statistical concepts do not mix.
CAUSAL
Spurious correlation
Randomization
Confounding / Effect
Instrument
Holding constant
Explanatory variables
STATISTICAL
Regression
Association / Independence
“Controlling for” / Conditioning
Odd and risk ratios
Collapsibility
2. No causes in – no causes out (Cartwright, 1989)
statistical assumptions + data
causal conclusions
causal assumptions
}
3. Causal assumptions cannot be expressed in the mathematical
language of standard statistics.
4.
FROM STATISTICAL TO CAUSAL ANALYSIS:
1. THE DIFFERENCES (CONT)
1. Causal and statistical concepts do not mix.
CAUSAL
Spurious correlation
Randomization
Confounding / Effect
Instrument
Holding constant
Explanatory variables
STATISTICAL
Regression
Association / Independence
“Controlling for” / Conditioning
Odd and risk ratios
Collapsibility
2. No causes in – no causes out (Cartwright, 1989)
statistical assumptions + data
causal conclusions
causal assumptions
}
3. Causal assumptions cannot be expressed in the mathematical
language of standard statistics.
4. Non-standard mathematics:
a) Structural equation models (Wright, 1920; Simon, 1960)
b) Counterfactuals (Neyman-Rubin (Yx), Lewis (x
Y))
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,…)
Structural model
Counterfactuals are
derived quantities
Subscripts modify the
distribution
X, Y, Z constrain Yx, Zy,… P(Yx=y) = Px(Y=y)
“SUPER” DISTRIBUTION
IN N-R MODEL
X
Y
Z
Yx=0
Yx=1
Xz=0
Xz=1
Xy=0
U
0
0
0
0
1
0
0
0
1
1
1
0
1
0
0
1
u1
u2
0
0
0
1
0
0
1
1
u3
1
0
0
1
0
0
1
0
u4
inconsistency:
Defines :
x = 0  Yx=0 = Y
Y = xY1 + (1-x) Y0
P * ( X , Y , Z ,...Yx , Z y ...Yxz , Z xy ,... ...)
P * (Yx  y | Z , X z )
Yx  X | Z y
TYPICAL INFERENCE
IN N-R MODEL
Find P*(Yx=y) given covariate Z,
P * (Yx  y )   P * (Yx  y | z ) P ( z )
z
Assume ignorability:
Yx  X | Z
Assume consistency:
X=x  Yx=Y
  P * (Yx  y | x, z ) P ( z )
z
  P * (Y  y | x, z ) P ( z )
z
  P ( y | x, z ) P ( z )
z
Problems:
Try it: X  Y  Z
?
1) Yx  X | Z judgmental & opaque
2) Is consistency the only connection between
X, Y and Yx?
THE STRUCTURAL MODEL
PARADIGM
Data
Joint
Distribution
Data
Generating
Model
Q(M)
(Aspects of M)
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.
FAMILIAR CAUSAL MODEL
ORACLE FOR MANIPILATION
X
Y
Z
INPUT
OUTPUT
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
STRUCTURAL MODELS AND
CAUSAL DIAGRAMS
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
q  b1 p  d1i  u1
p  b2q  d 2 w  u2
I
W
Q
P
U2
PAQ
STRUCTURAL MODELS AND
INTERVENTION
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
q  b1 p  d1i  u1
p  b2q  d 2 w  u2
U1
I
W
Q
P
U2
STRUCTURAL MODELS AND
INTERVENTION
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
q  b1 p  d1i  u1 U1
p  b2q  d 2 w  u2
p  p0
I
W
U2
Q
P
P = p0
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:
P(Yx  y, Z w  z ) 

u:Yx (u )  y,Z w (u )  z
P(u )
The super-distribution P* is derived from M.
Parsimonous, consistent, and transparent
AXIOMS OF CAUSAL
COUNTERFACTUALS
Yx (u )  y : Y would be y, had X been x (in state U = u)
1. Definiteness
x  X s.t. X y (u )  x
2. Uniqueness
( X y (u )  x) & ( X y (u )  x' )  x  x'
3. Effectiveness
X xw (u )  x
4. Composition
W x(u )  w  Yxw (u )  Yx (u )
5. Reversibility
(Yxw (u )  y & (Wxy (u )  w)  Yx (u )  y
GRAPHICAL – COUNTERFACTUALS
SYMBIOSIS
Every causal graph expresses counterfactuals
assumptions, e.g., X  Y  Z
1. Missing arrows Y  Z
Yx, z (u )  Yx (u )
2. Missing arcs
Yx  Z y
Y
Z
consistent, and readable from the graph.
Every theorem in SEM is a theorem in N-R,
and conversely.
STRUCTURAL ANALYSIS:
SOME USEFUL RESULTS
1. Complete formal semantics of counterfactuals
2. Transparent language for expressing assumptions
3. Complete solution to causal-effect identification
4. Legal responsibility (bounds)
5. Non-compliance (universal bounds)
6. Integration of data from diverse sources
7. Direct and Indirect effects,
8. Complete criterion for counterfactual testability
REGRESSION VS. STRUCTURAL EQUATIONS
(THE CONFUSION OF THE CENTURY)
Regression (claimless):
Y = ax + a1z1 + a2z2 + ... + akzk + y
a 
= E[Y | x,z] / x = Ryx z Y  y | X,Z
Structural (empirical, falsifiable):
Y = bx + b1z1 + b2z2 + ... + 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
REGRESSION VS. STRUCTURAL EQUATIONS
(THE CONFUSION OF THE CENTURY)
Regression (claimless):
Y = ax + a1z1 + a2z2 + ... + akzk + y
a 
= E[Y | x,z] / x = Ryx z Y  y | X,Z
Structural (empirical, falsifiable):
Y = bx + b1z1 + b2z2 + ... + 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
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.
Gx
G
Z1
Z1
Z2
Z3
Z2
Z3
Z4
X
Z
Z6
Z5
Y
Z4
X
Moreover, P(y | do(x)) =  P(y | x,z) P(z)
z
(“adjusting” for Z)
Z6
Z5
Y
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).
DETERMINING THE CAUSES OF EFFECTS
(The Attribution Problem)
•
•
Your Honor! My client (Mr. A) died BECAUSE
he used that drug.
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!
PN = P(? | A is dead, took the drug) > 0.50
THE PROBLEM
Semantical Problem:
1. 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.”
•
THE PROBLEM
Semantical Problem:
1. 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:
PN ( x, y )  P(Yx'  y ' | x, y )
Computable from M
THE PROBLEM
Semantical Problem:
1. 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.”
Analytical Problem:
2. Under what condition can PN(x,y) be learned from
statistical data, i.e., observational, experimental
and combined.
TYPICAL THEOREMS
(Tian and Pearl, 2000)
•
Bounds given combined nonexperimental and
experimental data
0


 1 
 P( y )  P( y ) 
 P( y' ) 
x'
x'
max 

PN

min



P( x,y )


 P( x,y ) 




•
Identifiability under monotonicity (Combined data)
P( y|x )  P( y|x' ) P( y|x' )  P( y x' )
PN 

P( y|x )
P( x,y )
corrected Excess-Risk-Ratio
CAN FREQUENCY DATA DECIDE
LEGAL RESPONSIBILITY?
Deaths (y)
Survivals (y)
•
•
•
•
Experimental
do(x) do(x)
16
14
984
986
1,000 1,000
Nonexperimental
x
x
2
28
998
972
1,000 1,000
Nonexperimental data: drug usage predicts longer life
Experimental data: drug has negligible effect on survival
Plaintiff: Mr. A is special.
1. He actually died
2. 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?
PN 
 P(Yx'  y' | x, y )  0.50
SOLUTION TO THE
ATTRIBUTION PROBLEM
•
•
WITH PROBABILITY ONE 1  P(yx | x,y)  1
Combined data tell more that each study alone
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)