Judea Pearl

1
THE MATHEMATICS OF CAUSE AND EFFECT

• Judea Pearl
• University of California
• Los Angeles

2
GENETIC MODELS (S. WRIGHT, 1920)
3
OUTLINE

• Lecture 1. Monday 330-530
• Why causal talk?
• Actions and Counterfactuals
• Identifying and bounding causal effects
• Policy Analysis
• Lecture 2. Tuesday 300-500
• Identifying and bounding probabilities of causes
• Attribution
• The Actual Cause
• Explanation
• References http//bayes.cs.ucla.edu/jp_home.html
• Slides transcripts
• CAUSALITY (forthcoming)

4
David Hume (17111776)
5
HUMES LEGACY

1. Analytical vs. empirical claims
2. Causal claims are empirical
3. All empirical claims originate from experience.

6
THE TWO RIDDLESOF CAUSATION

• What empirical evidence legitimizes a
  cause-effect connection?
• What inferences can be drawn from causal
  information? and how?

7
(No Transcript)
8
The Art ofCausal Mentoring
Easy, man! that hurts!
9
OLD RIDDLES IN NEW DRESS

• How should a robot acquire causal
• information from the environment?
• How should a robot process causal
• information received from its
• creator-programmer?

10
CAUSATION AS A PROGRAMMER'S NIGHTMARE

• Input
• If the grass is wet, then it rained
• if we break this bottle, the grass
• will get wet
• Output
• If we break this bottle, then it rained

11
CAUSATION AS APROGRAMMER'S NIGHTMARE (Cont.)
( Lin, 1995)

• Input
• A suitcase will open iff both
• locks are open.
• The right lock is open
• Query
• What if we open the left lock?
• Output
• The right lock might get closed.

12
THE BASIC PRINCIPLES
Causation encoding of behavior
under interventions Interventions surgeries
on mechanisms Mechanisms
stable functional relationships
equations graphs
13
WHAT'S IN A CAUSAL MODEL?
Oracle that assigns truth value to
causal sentences Action sentences B if
we do A. Counterfactuals ?B ? B if it were
A. Explanation B occurred because of
A. Optional with what probability?
14
CAUSAL MODELS WHY THEY ARE NEEDED
X
Y
Z
INPUT
OUTPUT
15
CAUSAL MODELS AT WORK(The impatient
firing-squad)
U (Court order)
C (Captain)
B (Riflemen)
A
D (Death)
16
CAUSAL MODELS AT WORK (Glossary)
U Court orders the execution C Captain
gives a signal A Rifleman-A shoots B Rifleman-B
shoots D Prisoner dies Functional Equality
(new symbol)
U
CU
C
AC
BC
B
A
DA?B
D
17
SENTENCES TO BE EVALUATED
U
S1. prediction ?A ? ?D S2. abduction ?D ?
?C S3. transduction A ? B S4. action
?C ? DA S5. counterfactual D ? D?A S6.
explanation Caused(A, D)
C
B
A
D
18
STANDARD MODEL FOR STANDARD QUERIES
S1. (prediction) If rifleman-A shot, the
prisoner is dead, A ? D S2. (abduction) If
the prisoner is alive, then the Captain did not
signal, ?D ? ?C S3. (transduction) If
rifleman-A shot, then B shot as well, A ? B
U
iff
C
iff
iff
B
A
?OR
D
19
WHY CAUSAL MODELS? GUIDE FOR SURGERY
S4. (action) If the captain gave no signal
and Mr. A decides to shoot, the prisoner will
die ?C ? DA, and B will not shoot ?C ?
?BA
U
C
B
A
D
20
WHY CAUSAL MODELS? GUIDE FOR SURGERY
S4. (action) If the captain gave no signal
and Mr. A decides to shoot, the prisoner will
die ?C ? DA, and B will not shoot ?C ?
?BA
U
C
B
A
D
21
MUTILATION IN SYMBOLIC CAUSAL MODELS
Model MA (Modify AC) (U) C U (C) A C
(A) B C (B) D A ? B (D) Facts
?C Conclusions ?
S4. (action) If the captain gave no signal and
A decides to shoot, the prisoner will die
and B will not shoot, ?C ? DA ?BA
22
MUTILATION IN SYMBOLIC CAUSAL MODELS
Model MA (Modify AC) (U) C U (C)
(A) B C (B) D A ? B (D) Facts
?C Conclusions ?
AC
S4. (action) If the captain gave no signal and
A decides to shoot, the prisoner will die
and B will not shoot, ?C ? DA ?BA
23
MUTILATION IN SYMBOLIC CAUSAL MODELS
Model MA (Modify AC) (U) C U (C) A
(A) B C (B) D A ? B (D) Facts
?C Conclusions A, D, ?B, ?U, ?C
AC
S4. (action) If the captain gave no signal and
A decides to shoot, the prisoner will die
and B will not shoot, ?C ? DA ?BA
24
3-STEPS TO COMPUTING COUNTERFACTUALS
S5. If the prisoner is dead, he would still be
dead if A had not shot. D?D?A
Abduction
Action
Prediction
U
TRUE
C
B
A
D
25
COMPUTING PROBABILITIES OF COUNTERFACTUALS
P(S5). The prisoner is dead. How likely is it
that he would be dead if A had not shot.
P(D?AD) ?
Abduction
Action
Prediction
U
C
B
A
D
26
SYMBOLIC EVALUATION OF COUNTERFACTUALS
Prove D ?D?A Combined Theory (U) C
U C U (C) ?A A C (A) B C B C
(B) D A ? B D A ? B (D) Facts
D Conclusions U, A, B, C, D, ?A, C, B, D
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PROBABILITY OF COUNTERFACTUALS THE TWIN
NETWORK
U
W
C
C
FALSE
?
B
B
A
A
TRUE
D
D
TRUE
P(Alive had A not shot A shot, Dead) P(?D)
in model ltM?A, P(u,wA,D)gt P(?DD) in
twin-network
28
CAUSAL MODEL (FORMAL)
M ltU, V, Fgt U - Background variables V -
Endogenous variables F - Set of functions U ? V
\Vi ?Vi vi fi (pai , ui ) Submodel Mx
ltU, V, Fxgt, representing do(x) Fx Replaces
equation for X with Xx Actions and
Counterfactuals Yx(u) Solution of Y in Mx
29
WHY COUNTERFACTUALS?
Action queries are triggered by (modifiable)
observations, demanding abductive step, i.e.,
counterfactual processing. E.g.,
Troubleshooting Observation The output is
low Action query Will the output get higher
if we replace the transistor? Counterfactua
l query Would the output be higher had the
transistor been replaced?
30
WHY CAUSALITY? FROM MECHANISMS TO MODALITY
Causality-free specification Causal
specification Prerequisite one-to-one
correspondence between variables and
mechanisms
action name
mechanism name
ramifications
direct-effects do(p)
ramifications
31
MID-STORY OUTLINE
Background From Hume to robotics Semantics and
principles Causal models, Surgeries, Actions
and Counterfactuals Applications I Evaluating
Actions and Plans from Data and
Theories Applications II Finding Explanations
and Single-event Causation
32
INTERVENTION AS SURGERY
Example Policy analysis
Model underlying data
Model for policy evaluation
Economic conditions
Economic conditions
Tax
Tax
Economic consequences
Economic consequences
33
PREDICTING THE EFFECTS OF POLICIES
1. Surgeon General (1964)
P (c do(s)) ? P (c s)
Smoking
Cancer
2. Tobacco Industry
Genotype (unobserved)
P (c do(s)) P (c)
Smoking
Cancer
3. Combined
P (c do(s)) noncomputable
Cancer
Smoking
34
PREDICTING THE EFFECTS OF POLICIES
1. Surgeon General (1964)
P (c do(s)) ? P (c s)
Smoking
Cancer
2. Tobacco Industry
Genotype (unobserved)
P (c do(s)) P (c)
Smoking
Cancer
3. Combined
P (c do(s)) noncomputable
Cancer
Smoking
35
PREDICTING THE EFFECTS OF POLICIES
1. Surgeon General (1964)
P (c do(s)) ? P (c s)
Smoking
Cancer
2. Tobacco Industry
Genotype (unobserved)
P (c do(s)) P (c)
Smoking
Cancer
3. Combined
P (c do(s)) noncomputable
Cancer
Smoking
4. Combined and refined
P (c do(s)) computable
36
The Science of Seeing
37
The Art of Doing
38
Combining Seeing and Doing
39
NEEDED ALGEBRA OF DOING
40
RULES OF CAUSAL CALCULUS
41
DERIVATION IN CAUSAL CALCULUS
Genotype (Unobserved)
Smoking
Tar
Cancer
Probability Axioms
P (c dos) ?t P (c dos, t) P (t dos)
Rule 2
?t P (c dos, dot) P (t dos)
Rule 2
?t P (c dos, dot) P (t s)
Rule 3
?t P (c dot) P (t s)
Probability Axioms
?s???t P (c dot, s?) P (s? dot) P(t s)
Rule 2
?s???t P (c t, s?) P (s? dot) P(t s)
Rule 3
?s? ?t P (c t, s?) P (s?) P(t s)
42
LEARNING TO ACT BY WATCHING OTHER ACTORS
U1
E.g., Process-control
Hidden dials
X1
U2
Control knobs
Z
X2
Visible dials
Y Output
Problem Find the effect of (do(x1), do(x2)) on
Y, from data on X1, Z, X2 and Y.
43
LEARNING TO ACT BY WATCHING OTHER ACTORS
Patients history
U1
Patients immune status
E.g., Drug-management (Pearl Robins, 1985)
X1
U2
Dosages Of Bactrim
Z
Episodes of PCP
X2
Y recovery/death
Solution P(ydo(x1), do(x2)) ?z P(yz, x1,
x2) P(zx1)
44
LEGAL ATTRIBUTION WHEN IS A DISEASE DUE TO
EXPOSURE?
Exposure to Radiation X
Enabling Factors
Q
AND
Other causes
U
OR
Y (Leukemia)
BUT-FOR criterion PNP(Yx? ? y X x,Y y)
gt 0.5 Q. When is PN identifiable from
P(x,y)? A. No confounding monotonicity PN
P(y x) ? P(y? x? ) / P(y x)
45
THE MATHEMATICS OF CAUSE AND EFFECT

• Judea Pearl
• University of California
• Los Angeles

46
OUTLINE

• Lecture 1. Monday 330-530
• Why causal talk?
• Actions and Counterfactuals
• Identifying and bounding causal effects
• Policy Analysis
• Lecture 2. Tuesday 300-500
• Identifying and bounding probabilities of causes
• Attribution
• The Actual Cause
• Explanation
• References http//bayes.cs.ucla.edu/jp_home.html
• Slides transcripts
• CAUSALITY (forthcoming)

47
APPLICATIONS-II

• Finding explanations for reported events
• Generating verbal explanations
• Understanding causal talk
• Formulating theories of causal thinking

48
Causal Explanation
She handed me the fruit and I ate
49
Causal Explanation
She handed me the fruit and I ate
The serpent deceived me, and I ate
50
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
51
PROBLEMS WITH THE COUNTERFACTUAL TEST
1. NECESSITY Ignores aspects of sufficiency
(Production) Fails in presence of other causes
(Overdetermination) 2. COARSENESS Ignores
structure of intervening mechanisms. Fails when
other causes are preempted (Preemption) SOLUTION
Supplement counterfactual test with Sustenance
52
THE IMPORTANCE OF SUFFICIENCY (PRODUCTION)
Match
Oxygen
AND
Fire
Observation Fire broke out. Question
Why is oxygen an awkward explanation?
Answer Because Oxygen is (usually) not
sufficient P(Oxygen is sufficient)
P(Match is lighted) low P(Match is sufficient)
P(Oxygen present) high
53
OVERDETERMINATION HOW THE COUNTERFACTUAL
TEST FAILS?
U (Court order)
C (Captain)
B (Riflemen)
A
D (Death)
Observation Dead prisoner with two
bullets. Query Was A a cause of death?
Answer Yes, A sustains D against B.
54
OVERDETERMINATION HOW THE SUSTENANCE TEST
SUCCEEDS?
U (Court order)
C (Captain)
B (Riflemen)
A
D (Death)
Observation Dead prisoner with two
bullets. Query Was A a cause of death?
Answer Yes, A sustains D against B.
55
NUANCES IN CAUSAL TALK
y depends on x (in u) X(u)x, Y(u)y, Yx?
(u)y? x can produce y (in u) X(u)x?, Y(u)y?,
Yx (u)y x sustains y relative to W X(u)x,
Y(u)y, Yx w (u)y, Yx? w? (u)y?
56
NUANCES IN CAUSAL TALK
x caused y, necessary for, responsible for, y
due to x, y attributed to x.
y depends on x (in u) X(u)x, Y(u)y, Yx?
(u)y? x can produce y (in u) X(u)x?, Y(u)y?,
Yx (u)y x sustains y relative to W X(u)x,
Y(u)y, Yxw (u)y, Yx?w? (u)y?
57
NUANCES IN CAUSAL TALK
y depends on x (in u) X(u)x, Y(u)y, Yx?
(u)y? x can produce y (in u) X(u)x?, Y(u)y?,
Yx (u)y x sustains y relative to W X(u)x,
Y(u)y, Yxw (u)y, Yx?w? (u)y?
x causes y, sufficient for, enables, triggers,
brings about, activates, responds
to, susceptible to.
58
NUANCES IN CAUSAL TALK
maintain, protect, uphold, keep up, back
up, prolong, support, rests on.
y depends on x (in u) X(u)x, Y(u)y, Yx?
(u)y? x can produce y (in u) X(u)x?, Y(u)y?,
Yx (u)y x sustains y relative to W X(u)x,
Y(u)y, Yxw (u)y, Yx? w? (u)y?
59
PREEMPTION HOW THE COUNTERFACTUAL TEST FAILS
Which switch is the actual cause of light? S1!
ON OFF
Switch-1
Light
Switch-2
Deceiving symmetry Light S1 ? S2
60
PREEMPTION HOW THE COUNTERFACTUAL TEST FAILS
Which switch is the actual cause of light? S1!
ON OFF
Switch-1
Light
Switch-2
Deceiving symmetry Light S1 ? S2
61
PREEMPTION HOW THE COUNTERFACTUAL TEST FAILS
Which switch is the actual cause of light? S1!
ON OFF
Switch-1
Light
Switch-2
Deceiving symmetry Light S1 ? S2
62
PREEMPTION HOW THE COUNTERFACTUAL TEST FAILS
Which switch is the actual cause of light? S1!
ON OFF
Switch-1
Light
Switch-2
Deceiving symmetry Light S1 ? S2
63
PREEMPTION HOW THE COUNTERFACTUAL TEST FAILS
Which switch is the actual cause of light? S1!
ON OFF
Switch-1
Light
Switch-2
Deceiving symmetry Light S1 ? S2
64
CAUSAL BEAM Locally sustaining sub-process
ACTUAL CAUSATION x is an actual cause of y in
scenario u, if x passes the following test
1. Construct a new model Beam(u, w ?) 1.1 In
each family, retain a subset of parents that
minimally sustains the child 1.2 Set the
other parents to some value w ? 2. Test if x
is necessary for y in Beam(u, w ?) for some w ?
65
THE DESERT TRAVELER (After Pat Suppes)
X
P
Enemy-2 Shoots canteen
Enemy -1 Poisons water
dehydration D
C cyanide intake
Y death
66
THE DESERT TRAVELER (The actual scenario)
Enemy-2 Shoots canteen
Enemy -1 Poisons water
dehydration D
C cyanide intake
Y death
67
THE DESERT TRAVELER (Constructing a causal beam)
Enemy-2 Shoots canteen
Enemy -1 Poisons water
? X ? P
dehydration D
C cyanide intake
Y death
68
THE DESERT TRAVELER (Constructing a causal beam)
Enemy-2 Shoots canteen
Enemy -1 Poisons water
C ? X
dehydration D
C cyanide intake
y death
69
THE DESERT TRAVELER (Constructing a causal beam)
Enemy-2 Shoots canteen
Enemy -1 Poisons water
C ? X
dehydration D
C cyanide intake
D ? C
y death
70
THE DESERT TRAVELER (The final beam)
Enemy-2 Shoots canteen
Enemy -1 Poisons water
C ? X
dehydration D
C cyanide intake
YD
YX
y death
71
THE ENIGMATIC DESERT TRAVELER (Uncertain
scenario)
U
P
time to first drink
P1
u
X1
Enemy-2 Shoots canteen
Enemy -1 Poisons water
dehydration D
C cyanide intake
y death
72
CAUSAL BEAM FOR THE DEHYDRATED TRAVELER
empty before drink
X 1
P 1
u 1
C 0
D 1
y 1
73
CAUSAL BEAM FOR THE POISONED TRAVELER
drink before empty
X 1
P 1
u 0
C 1
D 0
y 1
74
TEMPORAL PREEMPTION
Fire-1 is the actual cause of damage
Fire-1
House burned
Fire-2
Yet, Fire-1 fails the counterfactual test
75
TEMPORAL PREEMPTION AND DYNAMIC BEAMS
x
x
House
t
t
S(x,t) f S(x,t-1), S(x1, t-1), S(x-1,t-1)
76
DYNAMIC MODEL UNDER ACTION do(Fire-1),
do(Fire-2)
x
x
House
t
t
77
THE RESULTING SCENARIO
x
x
House
t
t
S(x,t) f S(x,t-1), S(x1, t-1), S(x-1,t-1)
78
THE DYNAMIC BEAM
x
x
House
t
t
Actual cause Fire-1
79
CONCLUSIONS
Development of Western science is based on two
great achievements the invention of the formal
logical system (in Euclidean geometry) by the
Greek philosophers, and the discovery of the
possibility to find out causal relationships by
systematic experiment (during the
Renaissance). A. Einstein, April 23, 1953
80
ACKNOWLEDGEMENT-I
Collaborators in Causality Alex Balke Moisés
Goldszmidt David Chickering Sander
Greenland Adnan Darwiche David Heckerman Rina
Dechter Jin Kim Hector Geffner Jamie
Robins David Galles Tom Verma
81
ACKNOWLEDGEMENT-II
Influential ideas S. Wright (1920) P.
Spirtes, C. Glymour T. Haavelmo (1943)
R. Scheines (1993) H. Simon (1953) P.
Nayak (1994) I.J. Good (1961) F. Lin
(1995) R. Strotz H. Wold (1963) D. Heckerman
D. Lewis (1973) R. Shachter (1995) R.
Reiter (1987) N. Hall (1998) Y. Shoham
(1988) J. Halpern (1998) M. Druzdzel D.
Michie (1998) H. Simon (1993)