A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to...
Transcript of A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to...
![Page 1: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/1.jpg)
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
![Page 2: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/2.jpg)
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
![Page 3: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/3.jpg)
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.
![Page 4: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/4.jpg)
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.
![Page 5: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/5.jpg)
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.
![Page 6: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/6.jpg)
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.
![Page 7: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/7.jpg)
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.
![Page 8: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/8.jpg)
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.
![Page 9: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/9.jpg)
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.
![Page 10: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/10.jpg)
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.
![Page 11: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/11.jpg)
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.
![Page 12: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/12.jpg)
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.
![Page 13: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/13.jpg)
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.
![Page 14: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/14.jpg)
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.
![Page 15: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/15.jpg)
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.
![Page 16: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/16.jpg)
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.
![Page 17: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/17.jpg)
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.
![Page 18: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/18.jpg)
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.
![Page 19: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/19.jpg)
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
![Page 20: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/20.jpg)
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
![Page 21: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/21.jpg)
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
![Page 22: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/22.jpg)
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
![Page 23: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/23.jpg)
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
![Page 24: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/24.jpg)
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.
![Page 25: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/25.jpg)
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.
![Page 26: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/26.jpg)
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.
![Page 27: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/27.jpg)
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
![Page 28: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/28.jpg)
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
![Page 29: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/29.jpg)
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
![Page 30: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/30.jpg)
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
![Page 31: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/31.jpg)
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
![Page 32: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/32.jpg)
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.
![Page 33: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/33.jpg)
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.
![Page 34: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/34.jpg)
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.
![Page 35: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/35.jpg)
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.)
![Page 36: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/36.jpg)
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
![Page 37: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/37.jpg)
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
![Page 38: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/38.jpg)
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.
![Page 39: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/39.jpg)
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.
![Page 40: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/40.jpg)
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.
![Page 41: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/41.jpg)
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.
![Page 42: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/42.jpg)
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.
![Page 43: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/43.jpg)
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.
![Page 44: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/44.jpg)
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
![Page 45: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/45.jpg)
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
![Page 46: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/46.jpg)
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.
![Page 47: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/47.jpg)
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.
![Page 48: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/48.jpg)
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.
![Page 49: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/49.jpg)
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.
![Page 50: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/50.jpg)
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.
![Page 51: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/51.jpg)
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)
![Page 52: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/52.jpg)
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)
![Page 53: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/53.jpg)
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)
![Page 54: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/54.jpg)
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)
![Page 55: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/55.jpg)
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)
![Page 56: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/56.jpg)
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)
![Page 57: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/57.jpg)
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)
![Page 58: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/58.jpg)
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)
![Page 59: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/59.jpg)
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
![Page 60: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/60.jpg)
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
![Page 61: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/61.jpg)
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
![Page 62: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/62.jpg)
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
![Page 63: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/63.jpg)
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
![Page 64: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/64.jpg)
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
![Page 65: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/65.jpg)
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
![Page 66: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/66.jpg)
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
![Page 67: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/67.jpg)
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
![Page 68: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/68.jpg)
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
![Page 69: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/69.jpg)
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
![Page 70: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/70.jpg)
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
![Page 71: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/71.jpg)
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
![Page 72: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/72.jpg)
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
![Page 73: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/73.jpg)
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
![Page 74: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/74.jpg)
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)
![Page 75: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/75.jpg)
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)
![Page 76: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/76.jpg)
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)
![Page 77: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/77.jpg)
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)
![Page 78: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/78.jpg)
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)
![Page 79: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/79.jpg)
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)
![Page 80: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/80.jpg)
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.
![Page 81: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/81.jpg)
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.
![Page 82: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/82.jpg)
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
![Page 83: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/83.jpg)
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
![Page 84: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/84.jpg)
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
![Page 85: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/85.jpg)
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
![Page 86: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/86.jpg)
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
![Page 87: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/87.jpg)
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
![Page 88: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/88.jpg)
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
![Page 89: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/89.jpg)
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).
![Page 90: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/90.jpg)
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∗.
![Page 91: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/91.jpg)
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∗.
![Page 92: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/92.jpg)
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∗.
![Page 93: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/93.jpg)
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∗.
![Page 94: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/94.jpg)
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∗.
![Page 95: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/95.jpg)
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!
![Page 96: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/96.jpg)
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?
![Page 97: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/97.jpg)
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?
![Page 98: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/98.jpg)
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?
![Page 99: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/99.jpg)
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?
![Page 100: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/100.jpg)
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?
![Page 101: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/101.jpg)
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?
![Page 102: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/102.jpg)
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?
![Page 103: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/103.jpg)
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∗.
![Page 104: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/104.jpg)
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∗.
![Page 105: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/105.jpg)
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
![Page 106: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/106.jpg)
A New CausalPower Theory
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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
![Page 107: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/107.jpg)
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
![Page 108: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/108.jpg)
A New CausalPower Theory
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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
![Page 109: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/109.jpg)
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
![Page 110: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/110.jpg)
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
![Page 111: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/111.jpg)
A New CausalPower Theory
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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?
![Page 112: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/112.jpg)
A New CausalPower Theory
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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?
![Page 113: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/113.jpg)
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?
![Page 114: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/114.jpg)
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?
![Page 115: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/115.jpg)
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?
![Page 116: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/116.jpg)
A New CausalPower Theory
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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.
![Page 117: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/117.jpg)
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.
![Page 118: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/118.jpg)
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.
![Page 119: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/119.jpg)
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.
![Page 120: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/120.jpg)
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.
![Page 121: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/121.jpg)
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.
![Page 122: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/122.jpg)
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.
![Page 123: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/123.jpg)
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.
![Page 124: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/124.jpg)
A New CausalPower Theory
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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
![Page 125: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/125.jpg)
A New CausalPower Theory
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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
![Page 126: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/126.jpg)
A New CausalPower Theory
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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.
![Page 127: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/127.jpg)
A New CausalPower Theory
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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)
![Page 128: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/128.jpg)
A New CausalPower Theory
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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.
![Page 129: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/129.jpg)
A New CausalPower Theory
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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
![Page 130: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/130.jpg)
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
![Page 131: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/131.jpg)
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
![Page 132: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/132.jpg)
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
![Page 133: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/133.jpg)
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
![Page 134: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/134.jpg)
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
![Page 135: A New Causal Power Theory - Univ of Kent 2008 · The power of a doctor’s advice to exercise to prevent heart attacks. A New Causal Power Theory 4/39 Causal Power Theory Wright’s](https://reader035.fdocuments.us/reader035/viewer/2022080718/5f78dd2baec3c5629754c8d5/html5/thumbnails/135.jpg)
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