Post on 27-Jul-2020
A New CausalPower Theory
1/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
A New Causal Power TheoryUniv of Kent
2008
Kevin Korb1 Erik Nyberg2
1Clayton School of ITMonash University
2History and Philosophy of ScienceUniversity of Melbourne
A New CausalPower Theory
2/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Contents
1 Causal Power Theory
2 Wright’s Theory
3 PC Theory
4 Causal Information
A New CausalPower Theory
3/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power Theories
• S Wright (1934) The method of path coefficients. Ann Math Stat, 5,161-215.
• IJ Good (1961) A causal calculus. BJPS, 11, 305-318.• P Cheng (1997) From covariation to causation. Psych Rev, 104,
367-405.• C Glymour & P Cheng (1998) Causal mechanism and probability.
Oaksford and Chater (Eds.) Rational models of cognition. Oxford.• C Glymour (2001) The Mind’s Arrows. MIT Press.• C Hitchcock (2001) The intransitivity of causation. JP, 98, 273-299.• E Hiddleston (2005) Causal powers. BJPS, 56, 27-59.• L Hope and K Korb (2005) An Information-theoretic causal power theory.
Australian AI Conference, pp. 805-811. Springer.
A New CausalPower Theory
4/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
What is causal power?
The power of some event to bring about(prevent) another event.
Examples• The power of anticoagulants to prevent death from
heart attack.• The power of exercise to prevent heart attacks.• The power of a doctor’s advice to exercise to bring
about exercise.• The power of a doctor’s advice to exercise to prevent
heart attacks.
A New CausalPower Theory
4/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
What is causal power?
The power of some event to bring about(prevent) another event.
Examples• The power of anticoagulants to prevent death from
heart attack.• The power of exercise to prevent heart attacks.• The power of a doctor’s advice to exercise to bring
about exercise.• The power of a doctor’s advice to exercise to prevent
heart attacks.
A New CausalPower Theory
4/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
What is causal power?
The power of some event to bring about(prevent) another event.
Examples• The power of anticoagulants to prevent death from
heart attack.• The power of exercise to prevent heart attacks.• The power of a doctor’s advice to exercise to bring
about exercise.• The power of a doctor’s advice to exercise to prevent
heart attacks.
A New CausalPower Theory
4/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
What is causal power?
The power of some event to bring about(prevent) another event.
Examples• The power of anticoagulants to prevent death from
heart attack.• The power of exercise to prevent heart attacks.• The power of a doctor’s advice to exercise to bring
about exercise.• The power of a doctor’s advice to exercise to prevent
heart attacks.
A New CausalPower Theory
4/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
What is causal power?
The power of some event to bring about(prevent) another event.
Examples• The power of anticoagulants to prevent death from
heart attack.• The power of exercise to prevent heart attacks.• The power of a doctor’s advice to exercise to bring
about exercise.• The power of a doctor’s advice to exercise to prevent
heart attacks.
A New CausalPower Theory
4/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
What is causal power?
The power of some event to bring about(prevent) another event.
Examples• The power of anticoagulants to prevent death from
heart attack.• The power of exercise to prevent heart attacks.• The power of a doctor’s advice to exercise to bring
about exercise.• The power of a doctor’s advice to exercise to prevent
heart attacks.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
5/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
One important point:
Causal power is always relative to a referenceclass.
• The power of the pill to prevent pregnancy• Amongst women• Amongst men
• The power of extra exercise to prevent heart attacks.• Amongst middle-aged couch potatoes• Amongst athletes• Amongst teenagers
Usually the reference class is implicit.
A New CausalPower Theory
6/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
We should like to develop an explicit quantitative measureof causal power, generalizing (improving on) our intuitivejudgments.
• Stochastic causality comes in degrees (“effect size”in medicine)
• Potentially allowing for precise judgments of causalattribution
• hence, the interest of cog psych
• Clarifying the explanatory import of causal Bayesiannetworks
A New CausalPower Theory
6/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
We should like to develop an explicit quantitative measureof causal power, generalizing (improving on) our intuitivejudgments.
• Stochastic causality comes in degrees (“effect size”in medicine)
• Potentially allowing for precise judgments of causalattribution
• hence, the interest of cog psych
• Clarifying the explanatory import of causal Bayesiannetworks
A New CausalPower Theory
6/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
We should like to develop an explicit quantitative measureof causal power, generalizing (improving on) our intuitivejudgments.
• Stochastic causality comes in degrees (“effect size”in medicine)
• Potentially allowing for precise judgments of causalattribution
• hence, the interest of cog psych
• Clarifying the explanatory import of causal Bayesiannetworks
A New CausalPower Theory
6/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
We should like to develop an explicit quantitative measureof causal power, generalizing (improving on) our intuitivejudgments.
• Stochastic causality comes in degrees (“effect size”in medicine)
• Potentially allowing for precise judgments of causalattribution
• hence, the interest of cog psych
• Clarifying the explanatory import of causal Bayesiannetworks
A New CausalPower Theory
6/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Power
We should like to develop an explicit quantitative measureof causal power, generalizing (improving on) our intuitivejudgments.
• Stochastic causality comes in degrees (“effect size”in medicine)
• Potentially allowing for precise judgments of causalattribution
• hence, the interest of cog psych
• Clarifying the explanatory import of causal Bayesiannetworks
A New CausalPower Theory
7/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
Most theories of causal power are based on binarynetworks (Cheng, Glymour, Hiddleston).
The first theory, Wright (1934), uses standardized linearGaussian models: path models.
Desideratum 1Causal power theory should apply to any kind of causalBayesian network – linear, binomial, multinomial.
A New CausalPower Theory
7/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
Most theories of causal power are based on binarynetworks (Cheng, Glymour, Hiddleston).
The first theory, Wright (1934), uses standardized linearGaussian models: path models.
Desideratum 1Causal power theory should apply to any kind of causalBayesian network – linear, binomial, multinomial.
A New CausalPower Theory
7/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
Most theories of causal power are based on binarynetworks (Cheng, Glymour, Hiddleston).
The first theory, Wright (1934), uses standardized linearGaussian models: path models.
Desideratum 1Causal power theory should apply to any kind of causalBayesian network – linear, binomial, multinomial.
A New CausalPower Theory
8/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
31
32
21p
p
p
X1
X2X3
r 1 2 31 12 r12 13 r13 r23 1
A New CausalPower Theory
9/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
Theorem (Explained Variation)Path coefficients are equal to the square root of thevariation in the child variable attributable to the parent.
I.e., ∑i
p2ji = 1
• As a consequence of standardization• Requires a residual term U with coefficient pju
A New CausalPower Theory
9/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
Theorem (Explained Variation)Path coefficients are equal to the square root of thevariation in the child variable attributable to the parent.
I.e., ∑i
p2ji = 1
• As a consequence of standardization• Requires a residual term U with coefficient pju
A New CausalPower Theory
9/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
Theorem (Explained Variation)Path coefficients are equal to the square root of thevariation in the child variable attributable to the parent.
I.e., ∑i
p2ji = 1
• As a consequence of standardization• Requires a residual term U with coefficient pju
A New CausalPower Theory
9/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Path Models
Theorem (Explained Variation)Path coefficients are equal to the square root of thevariation in the child variable attributable to the parent.
I.e., ∑i
p2ji = 1
• As a consequence of standardization• Requires a residual term U with coefficient pju
A New CausalPower Theory
10/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
Wright developed a graphical rule for relating (observed)correlations with path coefficients (i.e., relating probabilityand causality).
Fundamental idea: correlation results fromcausal influence along certain paths betweenvariables.
Definition (Admissible Path)
Φk is an admissible path between Xi and Xj iff it is anundirected path connecting Xi and Xj s.t. it does not goagainst the direction of an arc after having gone forward.
A New CausalPower Theory
10/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
Wright developed a graphical rule for relating (observed)correlations with path coefficients (i.e., relating probabilityand causality).
Fundamental idea: correlation results fromcausal influence along certain paths betweenvariables.
Definition (Admissible Path)
Φk is an admissible path between Xi and Xj iff it is anundirected path connecting Xi and Xj s.t. it does not goagainst the direction of an arc after having gone forward.
A New CausalPower Theory
10/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
Wright developed a graphical rule for relating (observed)correlations with path coefficients (i.e., relating probabilityand causality).
Fundamental idea: correlation results fromcausal influence along certain paths betweenvariables.
Definition (Admissible Path)
Φk is an admissible path between Xi and Xj iff it is anundirected path connecting Xi and Xj s.t. it does not goagainst the direction of an arc after having gone forward.
A New CausalPower Theory
11/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
This can be thought of as 3 rules in 1 for defining pathssupporting causal influence:
1 Directed chains support causal influence2 Common ancestors support causal influence
between descendants3 Common descendants don’t support causal
influence between ancestors
(This prefigures Pearl’s d-separation rules.)
A New CausalPower Theory
12/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
To assess the strength of causal influence along anadmissible path:
Definition (Valuation)The valuation of a path is
v(Φk ) =∏lm
plm for all Xm → Xl ∈ Φk
A New CausalPower Theory
12/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
To assess the strength of causal influence along anadmissible path:
Definition (Valuation)The valuation of a path is
v(Φk ) =∏lm
plm for all Xm → Xl ∈ Φk
A New CausalPower Theory
13/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
Theorem (Wright’s Decomposition Rule)The correlation rij between variables Xi and Xj , where Xiis an ancestor of Xj , can be rewritten as:
rij =∑
k
v(Φk )
where Φk is an admissible path between Xi and Xj andv(·) is a valuation of that path.
A New CausalPower Theory
14/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
This gives a direct relation between path coefficients andcorrelations:
r12 = p21
r13 = p31 + p21p32
r23 = p32 + p21p31
We can solve for the pij :
p21 = r12
p31 =r13 − r23r12
1− r212
p32 =r23 − r13r12
1− r212
Hence, we can parameterize (identify) any (recursive) pathmodel, given a correlation table.
A New CausalPower Theory
14/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
This gives a direct relation between path coefficients andcorrelations:
r12 = p21
r13 = p31 + p21p32
r23 = p32 + p21p31
We can solve for the pij :
p21 = r12
p31 =r13 − r23r12
1− r212
p32 =r23 − r13r12
1− r212
Hence, we can parameterize (identify) any (recursive) pathmodel, given a correlation table.
A New CausalPower Theory
14/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Decomposition Rule
This gives a direct relation between path coefficients andcorrelations:
r12 = p21
r13 = p31 + p21p32
r23 = p32 + p21p31
We can solve for the pij :
p21 = r12
p31 =r13 − r23r12
1− r212
p32 =r23 − r13r12
1− r212
Hence, we can parameterize (identify) any (recursive) pathmodel, given a correlation table.
A New CausalPower Theory
15/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Power Theory
Wright’s implicit causal power theory:The causal power of C for E is:
CP(C, E) =
∑k∏
lm plm for all Xm → Xl ∈ Φkfor all Φk = C → . . .→ E
NB: This is implicit in Wright’s treatment; Wright had noexplicit causal power theory.
A New CausalPower Theory
15/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Power Theory
Wright’s implicit causal power theory:The causal power of C for E is:
CP(C, E) =
∑k∏
lm plm for all Xm → Xl ∈ Φkfor all Φk = C → . . .→ E
NB: This is implicit in Wright’s treatment; Wright had noexplicit causal power theory.
A New CausalPower Theory
16/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Heart Attack Example
So: What is the causal power of BP for HA?
Note:• Backpath BP← X→ HA• Messy interaction btw BP and X upon HA
A New CausalPower Theory
17/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Heart Attack Example
Consider the linear approximation (dropping the messyinteraction):
BP at 50 Heart Attack
by 60
−0.8 −0.2
0.4
eXercise at 40
rBP,HA = pBP,X pHA,X + pHA,BP = 0.56
The Wright causal power of BP for HA• Discounts the backpath BP← X→ HA• Equals 0.4
A New CausalPower Theory
18/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Power Theory
• Relates variables C and E , not their values• To relate values, we should need to discretize
variable ranges in some way
• Wright’s theory has been very successful• Wright’s theory is compatible with current Bayesian
network theory
Desideratum 2Causal power theory should generalize Wright’s powertheory.
A New CausalPower Theory
18/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Power Theory
• Relates variables C and E , not their values• To relate values, we should need to discretize
variable ranges in some way
• Wright’s theory has been very successful• Wright’s theory is compatible with current Bayesian
network theory
Desideratum 2Causal power theory should generalize Wright’s powertheory.
A New CausalPower Theory
18/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Power Theory
• Relates variables C and E , not their values• To relate values, we should need to discretize
variable ranges in some way
• Wright’s theory has been very successful• Wright’s theory is compatible with current Bayesian
network theory
Desideratum 2Causal power theory should generalize Wright’s powertheory.
A New CausalPower Theory
18/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Power Theory
• Relates variables C and E , not their values• To relate values, we should need to discretize
variable ranges in some way
• Wright’s theory has been very successful• Wright’s theory is compatible with current Bayesian
network theory
Desideratum 2Causal power theory should generalize Wright’s powertheory.
A New CausalPower Theory
18/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Wright’s Power Theory
• Relates variables C and E , not their values• To relate values, we should need to discretize
variable ranges in some way
• Wright’s theory has been very successful• Wright’s theory is compatible with current Bayesian
network theory
Desideratum 2Causal power theory should generalize Wright’s powertheory.
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
19/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Modern Causal Power TheoryCheng & Glymour
The Cheng (1997) and Glymour & Cheng (1998) PCTheory applies to binary variables taking particularvalues, C = c and E = e, given assumptions:
• ∃ a direct causal connection C → E• C is independent of any other cause of E• C does not interact with any other cause of E• Probabilistic relevance:
∆P = p(e|c)− p(e|¬c) 6= 0• Spurious causes must be eliminated
• e.g., replaced by common causes
(Echoing Salmon on SR explanation andSuppes on probabilistic causation)
A New CausalPower Theory
20/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
“Power PC” Theory
Definition (Causal Power)For positive ∆P (generative cause), the power of c tobring about e:
pc =∆P
1− P(e|¬c)
Idea: ∆P directly is not a fair measure of pc
• since there is a background rate P(e|¬c)
• ∆P should be relativized to the remainder– those cases that would have been ¬e but for c
A New CausalPower Theory
20/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
“Power PC” Theory
Definition (Causal Power)For positive ∆P (generative cause), the power of c tobring about e:
pc =∆P
1− P(e|¬c)
Idea: ∆P directly is not a fair measure of pc
• since there is a background rate P(e|¬c)
• ∆P should be relativized to the remainder– those cases that would have been ¬e but for c
A New CausalPower Theory
20/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
“Power PC” Theory
Definition (Causal Power)For positive ∆P (generative cause), the power of c tobring about e:
pc =∆P
1− P(e|¬c)
Idea: ∆P directly is not a fair measure of pc
• since there is a background rate P(e|¬c)
• ∆P should be relativized to the remainder– those cases that would have been ¬e but for c
A New CausalPower Theory
20/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
“Power PC” Theory
Definition (Causal Power)For positive ∆P (generative cause), the power of c tobring about e:
pc =∆P
1− P(e|¬c)
Idea: ∆P directly is not a fair measure of pc
• since there is a background rate P(e|¬c)
• ∆P should be relativized to the remainder– those cases that would have been ¬e but for c
A New CausalPower Theory
21/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Definition (Preventive Causal Power)For negative ∆P (preventive cause), the power of c tostop e:
pc =−∆P
P(e|¬c)
Symmetrically• there is a background rate of failure to reach e,
P(¬e|¬c) = 1− P(e|¬c)
• so −∆P should be measured relative to theremainder
A New CausalPower Theory
21/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Definition (Preventive Causal Power)For negative ∆P (preventive cause), the power of c tostop e:
pc =−∆P
P(e|¬c)
Symmetrically• there is a background rate of failure to reach e,
P(¬e|¬c) = 1− P(e|¬c)
• so −∆P should be measured relative to theremainder
A New CausalPower Theory
21/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Definition (Preventive Causal Power)For negative ∆P (preventive cause), the power of c tostop e:
pc =−∆P
P(e|¬c)
Symmetrically• there is a background rate of failure to reach e,
P(¬e|¬c) = 1− P(e|¬c)
• so −∆P should be measured relative to theremainder
A New CausalPower Theory
21/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Definition (Preventive Causal Power)For negative ∆P (preventive cause), the power of c tostop e:
pc =−∆P
P(e|¬c)
Symmetrically• there is a background rate of failure to reach e,
P(¬e|¬c) = 1− P(e|¬c)
• so −∆P should be measured relative to theremainder
A New CausalPower Theory
22/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Consider the noisy-OR approximation:
Reconstruct variables as binary; delete arc between X and BP; eliminatemessy interaction btw X and BPThen:
∆P = P(HA|BP)− P(HA|¬BP) = 0.195
pc =∆P
1− P(HA|¬BP)= 0.20
The prob that high BP will kill someone, given survival o/w relative to the model
A New CausalPower Theory
22/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Consider the noisy-OR approximation:
Reconstruct variables as binary; delete arc between X and BP; eliminatemessy interaction btw X and BPThen:
∆P = P(HA|BP)− P(HA|¬BP) = 0.195
pc =∆P
1− P(HA|¬BP)= 0.20
The prob that high BP will kill someone, given survival o/w relative to the model
A New CausalPower Theory
22/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Consider the noisy-OR approximation:
Reconstruct variables as binary; delete arc between X and BP; eliminatemessy interaction btw X and BPThen:
∆P = P(HA|BP)− P(HA|¬BP) = 0.195
pc =∆P
1− P(HA|¬BP)= 0.20
The prob that high BP will kill someone, given survival o/w relative to the model
A New CausalPower Theory
22/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Consider the noisy-OR approximation:
Reconstruct variables as binary; delete arc between X and BP; eliminatemessy interaction btw X and BPThen:
∆P = P(HA|BP)− P(HA|¬BP) = 0.195
pc =∆P
1− P(HA|¬BP)= 0.20
The prob that high BP will kill someone, given survival o/w relative to the model
A New CausalPower Theory
22/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Consider the noisy-OR approximation:
Reconstruct variables as binary; delete arc between X and BP; eliminatemessy interaction btw X and BPThen:
∆P = P(HA|BP)− P(HA|¬BP) = 0.195
pc =∆P
1− P(HA|¬BP)= 0.20
The prob that high BP will kill someone, given survival o/w relative to the model
A New CausalPower Theory
22/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Consider the noisy-OR approximation:
Reconstruct variables as binary; delete arc between X and BP; eliminatemessy interaction btw X and BPThen:
∆P = P(HA|BP)− P(HA|¬BP) = 0.195
pc =∆P
1− P(HA|¬BP)= 0.20
The prob that high BP will kill someone, given survival o/w relative to the model
A New CausalPower Theory
22/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory
Consider the noisy-OR approximation:
Reconstruct variables as binary; delete arc between X and BP; eliminatemessy interaction btw X and BPThen:
∆P = P(HA|BP)− P(HA|¬BP) = 0.195
pc =∆P
1− P(HA|¬BP)= 0.20
The prob that high BP will kill someone, given survival o/w relative to the model
A New CausalPower Theory
23/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory Limitations
• Binary variables only• Power measured only between values of variables,
not btw variables themselves (as with Wright) – weshould like both
• Glymour (2001) shows that PC Theory is limited to“noisy-OR” relations in Bayesian networks⇒ Non-interactive and transitive causes only
But we know, for example, causality is nottransitive! (Finesteride, Hesslow’s example)
A New CausalPower Theory
23/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory Limitations
• Binary variables only• Power measured only between values of variables,
not btw variables themselves (as with Wright) – weshould like both
• Glymour (2001) shows that PC Theory is limited to“noisy-OR” relations in Bayesian networks⇒ Non-interactive and transitive causes only
But we know, for example, causality is nottransitive! (Finesteride, Hesslow’s example)
A New CausalPower Theory
23/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory Limitations
• Binary variables only• Power measured only between values of variables,
not btw variables themselves (as with Wright) – weshould like both
• Glymour (2001) shows that PC Theory is limited to“noisy-OR” relations in Bayesian networks⇒ Non-interactive and transitive causes only
But we know, for example, causality is nottransitive! (Finesteride, Hesslow’s example)
A New CausalPower Theory
23/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory Limitations
• Binary variables only• Power measured only between values of variables,
not btw variables themselves (as with Wright) – weshould like both
• Glymour (2001) shows that PC Theory is limited to“noisy-OR” relations in Bayesian networks⇒ Non-interactive and transitive causes only
But we know, for example, causality is nottransitive! (Finesteride, Hesslow’s example)
A New CausalPower Theory
23/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory Limitations
• Binary variables only• Power measured only between values of variables,
not btw variables themselves (as with Wright) – weshould like both
• Glymour (2001) shows that PC Theory is limited to“noisy-OR” relations in Bayesian networks⇒ Non-interactive and transitive causes only
But we know, for example, causality is nottransitive! (Finesteride, Hesslow’s example)
A New CausalPower Theory
23/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theory Limitations
• Binary variables only• Power measured only between values of variables,
not btw variables themselves (as with Wright) – weshould like both
• Glymour (2001) shows that PC Theory is limited to“noisy-OR” relations in Bayesian networks⇒ Non-interactive and transitive causes only
But we know, for example, causality is nottransitive! (Finesteride, Hesslow’s example)
A New CausalPower Theory
24/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Desiderata
Desideratum 3Causal power theory should apply both to variables andtheir values.
Desideratum 4Causal power theory should allow for non-transitive andinteractive relations.
A New CausalPower Theory
24/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Desiderata
Desideratum 3Causal power theory should apply both to variables andtheir values.
Desideratum 4Causal power theory should allow for non-transitive andinteractive relations.
A New CausalPower Theory
25/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Mutual Information
ConsiderP(c, e)
P(c)P(e)
The deviation of the joint distribution from independence
logP(c, e)
P(c)P(e)
Generalize:
MI(C, E) =dfX
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
• Mutual Information• Expected info about C given E , E given C
⇒ symmetric
• The standard measure of prob dependence
A New CausalPower Theory
25/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Mutual Information
ConsiderP(c, e)
P(c)P(e)
The deviation of the joint distribution from independence
logP(c, e)
P(c)P(e)
Generalize:
MI(C, E) =dfX
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
• Mutual Information• Expected info about C given E , E given C
⇒ symmetric
• The standard measure of prob dependence
A New CausalPower Theory
25/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Mutual Information
ConsiderP(c, e)
P(c)P(e)
The deviation of the joint distribution from independence
logP(c, e)
P(c)P(e)
Generalize:
MI(C, E) =dfX
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
• Mutual Information• Expected info about C given E , E given C
⇒ symmetric
• The standard measure of prob dependence
A New CausalPower Theory
25/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Mutual Information
ConsiderP(c, e)
P(c)P(e)
The deviation of the joint distribution from independence
logP(c, e)
P(c)P(e)
Generalize:
MI(C, E) =dfX
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
• Mutual Information• Expected info about C given E , E given C
⇒ symmetric
• The standard measure of prob dependence
A New CausalPower Theory
25/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Mutual Information
ConsiderP(c, e)
P(c)P(e)
The deviation of the joint distribution from independence
logP(c, e)
P(c)P(e)
Generalize:
MI(C, E) =dfX
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
• Mutual Information• Expected info about C given E , E given C
⇒ symmetric
• The standard measure of prob dependence
A New CausalPower Theory
25/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Mutual Information
ConsiderP(c, e)
P(c)P(e)
The deviation of the joint distribution from independence
logP(c, e)
P(c)P(e)
Generalize:
MI(C, E) =dfX
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
• Mutual Information• Expected info about C given E , E given C
⇒ symmetric
• The standard measure of prob dependence
A New CausalPower Theory
25/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Mutual Information
ConsiderP(c, e)
P(c)P(e)
The deviation of the joint distribution from independence
logP(c, e)
P(c)P(e)
Generalize:
MI(C, E) =dfX
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
• Mutual Information• Expected info about C given E , E given C
⇒ symmetric
• The standard measure of prob dependence
A New CausalPower Theory
26/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Fisher Model
Smoking Cancer
Gene
MI travels up and down any Wrightian path, includingback paths; causal influences clearly don’t (outside ofEPR problems).
A New CausalPower Theory
27/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Intervention
An intervention upon V ∈ g• Alters the distribution over V in g• From outside the system, outside g
Causal Bayesian networks are ideal forrepresenting interventions, augmenting g byadding an intervention variable I, yielding theaugmented g∗.
A New CausalPower Theory
27/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Intervention
An intervention upon V ∈ g• Alters the distribution over V in g• From outside the system, outside g
Causal Bayesian networks are ideal forrepresenting interventions, augmenting g byadding an intervention variable I, yielding theaugmented g∗.
A New CausalPower Theory
27/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Intervention
An intervention upon V ∈ g• Alters the distribution over V in g• From outside the system, outside g
Causal Bayesian networks are ideal forrepresenting interventions, augmenting g byadding an intervention variable I, yielding theaugmented g∗.
A New CausalPower Theory
27/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Intervention
An intervention upon V ∈ g• Alters the distribution over V in g• From outside the system, outside g
Causal Bayesian networks are ideal forrepresenting interventions, augmenting g byadding an intervention variable I, yielding theaugmented g∗.
A New CausalPower Theory
27/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Intervention
An intervention upon V ∈ g• Alters the distribution over V in g• From outside the system, outside g
Causal Bayesian networks are ideal forrepresenting interventions, augmenting g byadding an intervention variable I, yielding theaugmented g∗.
A New CausalPower Theory
28/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Fisher Model II
We shall use perfect (overwhelming) interventions tomeasure causal power
Smoking Cancer
GeneIntervention
which was, of course, Fisher’s idea!
A New CausalPower Theory
29/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal InformationIdea
Use MI, but asymmetrically• By first intervening perfectly upon C and only then
measuring MIWe get the asymmetrical dependence of E upon C, whenC is set to a fixed distribution.
This automates Wright’s power theory,• via Bayesian net tools• extending it automatically to all BNs
Remaining problem: which of the ℵ1distributions should we choose for C?
A New CausalPower Theory
29/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal InformationIdea
Use MI, but asymmetrically• By first intervening perfectly upon C and only then
measuring MIWe get the asymmetrical dependence of E upon C, whenC is set to a fixed distribution.
This automates Wright’s power theory,• via Bayesian net tools• extending it automatically to all BNs
Remaining problem: which of the ℵ1distributions should we choose for C?
A New CausalPower Theory
29/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal InformationIdea
Use MI, but asymmetrically• By first intervening perfectly upon C and only then
measuring MIWe get the asymmetrical dependence of E upon C, whenC is set to a fixed distribution.
This automates Wright’s power theory,• via Bayesian net tools• extending it automatically to all BNs
Remaining problem: which of the ℵ1distributions should we choose for C?
A New CausalPower Theory
29/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal InformationIdea
Use MI, but asymmetrically• By first intervening perfectly upon C and only then
measuring MIWe get the asymmetrical dependence of E upon C, whenC is set to a fixed distribution.
This automates Wright’s power theory,• via Bayesian net tools• extending it automatically to all BNs
Remaining problem: which of the ℵ1distributions should we choose for C?
A New CausalPower Theory
29/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal InformationIdea
Use MI, but asymmetrically• By first intervening perfectly upon C and only then
measuring MIWe get the asymmetrical dependence of E upon C, whenC is set to a fixed distribution.
This automates Wright’s power theory,• via Bayesian net tools• extending it automatically to all BNs
Remaining problem: which of the ℵ1distributions should we choose for C?
A New CausalPower Theory
29/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal InformationIdea
Use MI, but asymmetrically• By first intervening perfectly upon C and only then
measuring MIWe get the asymmetrical dependence of E upon C, whenC is set to a fixed distribution.
This automates Wright’s power theory,• via Bayesian net tools• extending it automatically to all BNs
Remaining problem: which of the ℵ1distributions should we choose for C?
A New CausalPower Theory
29/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal InformationIdea
Use MI, but asymmetrically• By first intervening perfectly upon C and only then
measuring MIWe get the asymmetrical dependence of E upon C, whenC is set to a fixed distribution.
This automates Wright’s power theory,• via Bayesian net tools• extending it automatically to all BNs
Remaining problem: which of the ℵ1distributions should we choose for C?
A New CausalPower Theory
30/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Information
Definition (Causal information (CI))Causal information between a cause C and an effect E inthe causal model g is
CI(C, E) =∑
c∈C,e∈E
p(c)p(e|c) logp(e|c)
p(e)
between the two variables in the augmented model g∗.
• This is precisely MI between C and E in theaugmented model g∗.
A New CausalPower Theory
30/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Causal Information
Definition (Causal information (CI))Causal information between a cause C and an effect E inthe causal model g is
CI(C, E) =∑
c∈C,e∈E
p(c)p(e|c) logp(e|c)
p(e)
between the two variables in the augmented model g∗.
• This is precisely MI between C and E in theaugmented model g∗.
A New CausalPower Theory
31/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Many-FlavoredCausal Information
Definition (CI various)
1 CI(C, e) =∑
c∈C p(c)p(e|c) log p(e|c)p(e)
2 CI(c, E) =∑
e∈E p(e|c) log p(e|c)p(e)
3 Causal Power: CI(c, e) = p(e|c) log p(e|c)p(e)
always measured in the augmented g∗.
1 Causal influence of C on E = e2 Causal influence of C = c on E3 Causal power of c to bring about e
A New CausalPower Theory
31/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Many-FlavoredCausal Information
Definition (CI various)
1 CI(C, e) =∑
c∈C p(c)p(e|c) log p(e|c)p(e)
2 CI(c, E) =∑
e∈E p(e|c) log p(e|c)p(e)
3 Causal Power: CI(c, e) = p(e|c) log p(e|c)p(e)
always measured in the augmented g∗.
1 Causal influence of C on E = e2 Causal influence of C = c on E3 Causal power of c to bring about e
A New CausalPower Theory
31/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Many-FlavoredCausal Information
Definition (CI various)
1 CI(C, e) =∑
c∈C p(c)p(e|c) log p(e|c)p(e)
2 CI(c, E) =∑
e∈E p(e|c) log p(e|c)p(e)
3 Causal Power: CI(c, e) = p(e|c) log p(e|c)p(e)
always measured in the augmented g∗.
1 Causal influence of C on E = e2 Causal influence of C = c on E3 Causal power of c to bring about e
A New CausalPower Theory
31/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Many-FlavoredCausal Information
Definition (CI various)
1 CI(C, e) =∑
c∈C p(c)p(e|c) log p(e|c)p(e)
2 CI(c, E) =∑
e∈E p(e|c) log p(e|c)p(e)
3 Causal Power: CI(c, e) = p(e|c) log p(e|c)p(e)
always measured in the augmented g∗.
1 Causal influence of C on E = e2 Causal influence of C = c on E3 Causal power of c to bring about e
A New CausalPower Theory
31/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Many-FlavoredCausal Information
Definition (CI various)
1 CI(C, e) =∑
c∈C p(c)p(e|c) log p(e|c)p(e)
2 CI(c, E) =∑
e∈E p(e|c) log p(e|c)p(e)
3 Causal Power: CI(c, e) = p(e|c) log p(e|c)p(e)
always measured in the augmented g∗.
1 Causal influence of C on E = e2 Causal influence of C = c on E3 Causal power of c to bring about e
A New CausalPower Theory
31/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Many-FlavoredCausal Information
Definition (CI various)
1 CI(C, e) =∑
c∈C p(c)p(e|c) log p(e|c)p(e)
2 CI(c, E) =∑
e∈E p(e|c) log p(e|c)p(e)
3 Causal Power: CI(c, e) = p(e|c) log p(e|c)p(e)
always measured in the augmented g∗.
1 Causal influence of C on E = e2 Causal influence of C = c on E3 Causal power of c to bring about e
A New CausalPower Theory
32/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Example Questions
Questions related to the various CI measures:
C for E :How much do heart attack outcomes depend uponBP?
C for e:How many heart attack deaths are due to BP?
c for E :How would heat attack outcomes vary givenlowered BP?
c for e:How many lives would be saved by interventions tolower BP?
A New CausalPower Theory
32/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Example Questions
Questions related to the various CI measures:
C for E :How much do heart attack outcomes depend uponBP?
C for e:How many heart attack deaths are due to BP?
c for E :How would heat attack outcomes vary givenlowered BP?
c for e:How many lives would be saved by interventions tolower BP?
A New CausalPower Theory
32/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Example Questions
Questions related to the various CI measures:
C for E :How much do heart attack outcomes depend uponBP?
C for e:How many heart attack deaths are due to BP?
c for E :How would heat attack outcomes vary givenlowered BP?
c for e:How many lives would be saved by interventions tolower BP?
A New CausalPower Theory
32/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Example Questions
Questions related to the various CI measures:
C for E :How much do heart attack outcomes depend uponBP?
C for e:How many heart attack deaths are due to BP?
c for E :How would heat attack outcomes vary givenlowered BP?
c for e:How many lives would be saved by interventions tolower BP?
A New CausalPower Theory
32/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Example Questions
Questions related to the various CI measures:
C for E :How much do heart attack outcomes depend uponBP?
C for e:How many heart attack deaths are due to BP?
c for E :How would heat attack outcomes vary givenlowered BP?
c for e:How many lives would be saved by interventions tolower BP?
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
33/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
ℵ1 Intervention Distributions
• Original p(C)
Given the actual distribution of BP, how does BPinfluence heart attack? (E.g., Swedes vsnon-Swedes)
• Uniform p(C)
As in randomized experimental designs
• Maximizing p(C)
What is the greatest possible influence of C forE? How strongly could lowering BP impact onheart attack outcomes?
The latter two provide a kind of standard baseline forcomparing causal powers.
A New CausalPower Theory
34/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Heart Attack ExampleMI vs CI
MI(BP, HA) =X
c∈C,e∈E
P(c, e) logP(c, e)
P(c)P(e)
= 0.28
CI(BP, HA) = 0.13
• The difference is due to the interventional elimination of the backpaththrough X
A New CausalPower Theory
35/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Heart Attack ExampleCI causal power
Two CI causal powers for fatal heart attack:
CI(c, e) = p(e|c) logp(e|c)
p(e)
• CI(high BP, fatal HA) = 0.23 log 0.230.0679 = 0.405
• CI(low BP, fatal HA) = 0.052 log 0.0520.0679 = −0.02
A New CausalPower Theory
36/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Heart Attack ExampleCheng
What happens to Cheng’s PC Theory when we apply it to theoriginal model?
The reintroduction of backpath and interaction
• pc = ∆P/[1− P(HA|¬BP)] = 0.16– a decline of 20%
This shows significant errors in attempting to apply PC Theory.
A New CausalPower Theory
37/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
PC Theories
Variables Structures CausalityWright Linear Open Transitive
Cheng/Glymour BinaryNoisy-ORIsolatedCauses
Transitive
CI Various OpenVarious(Interactions,thresholds)
A New CausalPower Theory
38/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
Desiderata
1 Causal power theory should apply to any kind ofcausal Bayesian network – linear, binomial,multinomial.
2 Causal power theory should generalize Wright’spower theory.
3 Causal power theory should apply both to variablesand their values.
4 Causal power theory should allow for non-transitiveand interactive relations.
A New CausalPower Theory
39/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
CI Summary
• CI reports the expected code length needed to reportthe value of E given the value of C in g∗
• This can be converted back into the language ofprobabilities
• CI satisfies all of our desiderata, unlike any knownalternative
• CI can summarize the explanatory import ofhypothetical causes, making causal BNs intelligible
• CI can be applied to test theories of causal attribution• CI can be applied to test theories of token causation
A New CausalPower Theory
39/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
CI Summary
• CI reports the expected code length needed to reportthe value of E given the value of C in g∗
• This can be converted back into the language ofprobabilities
• CI satisfies all of our desiderata, unlike any knownalternative
• CI can summarize the explanatory import ofhypothetical causes, making causal BNs intelligible
• CI can be applied to test theories of causal attribution• CI can be applied to test theories of token causation
A New CausalPower Theory
39/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
CI Summary
• CI reports the expected code length needed to reportthe value of E given the value of C in g∗
• This can be converted back into the language ofprobabilities
• CI satisfies all of our desiderata, unlike any knownalternative
• CI can summarize the explanatory import ofhypothetical causes, making causal BNs intelligible
• CI can be applied to test theories of causal attribution• CI can be applied to test theories of token causation
A New CausalPower Theory
39/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
CI Summary
• CI reports the expected code length needed to reportthe value of E given the value of C in g∗
• This can be converted back into the language ofprobabilities
• CI satisfies all of our desiderata, unlike any knownalternative
• CI can summarize the explanatory import ofhypothetical causes, making causal BNs intelligible
• CI can be applied to test theories of causal attribution• CI can be applied to test theories of token causation
A New CausalPower Theory
39/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
CI Summary
• CI reports the expected code length needed to reportthe value of E given the value of C in g∗
• This can be converted back into the language ofprobabilities
• CI satisfies all of our desiderata, unlike any knownalternative
• CI can summarize the explanatory import ofhypothetical causes, making causal BNs intelligible
• CI can be applied to test theories of causal attribution• CI can be applied to test theories of token causation
A New CausalPower Theory
39/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
CI Summary
• CI reports the expected code length needed to reportthe value of E given the value of C in g∗
• This can be converted back into the language ofprobabilities
• CI satisfies all of our desiderata, unlike any knownalternative
• CI can summarize the explanatory import ofhypothetical causes, making causal BNs intelligible
• CI can be applied to test theories of causal attribution• CI can be applied to test theories of token causation
A New CausalPower Theory
39/39
Causal PowerTheory
Wright’s Theory
PC Theory
Causal Information
CI Summary
• CI reports the expected code length needed to reportthe value of E given the value of C in g∗
• This can be converted back into the language ofprobabilities
• CI satisfies all of our desiderata, unlike any knownalternative
• CI can summarize the explanatory import ofhypothetical causes, making causal BNs intelligible
• CI can be applied to test theories of causal attribution• CI can be applied to test theories of token causation