when are graphical causal models not good models? capits 2008 jan lemeire september 12 th 2008
TRANSCRIPT
![Page 1: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/1.jpg)
When are Graphical Causal Models not Good Models?
CAPITS 2008
Jan LemeireSeptember 12th 2008
![Page 2: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/2.jpg)
Pag.Jan Lemeire / 222When are Causal Models not Good Models?
Overview
The Kolmogorov Minimal Sufficient Statistic (KMSS).
Bayesian Networks as Minimal Descriptions of Probability Distributions.
When the minimal Bayesian network is the KMSS.
When the minimal Bayesian network is NOT the KMSS.
![Page 3: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/3.jpg)
Pag.Jan Lemeire / 223When are Causal Models not Good Models?
Overview
The Kolmogorov Minimal Sufficient Statistic (KMSS).
Bayesian Networks as Minimal Descriptions of Probability Distributions.
When the minimal Bayesian network is the KMSS.
When the minimal Bayesian network is NOT the KMSS.
![Page 4: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/4.jpg)
Pag.Jan Lemeire / 224When are Causal Models not Good Models?
Randomness versus Regularity
001001001001001001001001001001001001001
011000110101101010111001001101000101110Random string: incompressible, maximal complexityBut: it is no meaningful information, only accidental
information
Regular string: compressible, low complexity
repetition 12 times, 001regularities randomness
Model that minimally describes regularities (qualitative props)= Kolmogorov Minimal Sufficient Statistic (KMSS)
![Page 5: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/5.jpg)
Pag.Jan Lemeire / 225When are Causal Models not Good Models?
Overview
The Kolmogorov Minimal Sufficient Statistic (KMSS).
Bayesian Networks as Minimal Descriptions of Probability Distributions.
When the minimal Bayesian network is the KMSS.
When the minimal Bayesian network is NOT the KMSS.
![Page 6: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/6.jpg)
Pag.Jan Lemeire / 226When are Causal Models not Good Models?
Description of Probability Distributionswith Bayesian networks
Multiple Bayesian networks select minimal one
Joint Probability Distribution
P(A, B, C, D, E)= C
E
B
A
D
GRAPH
P(A)P(B|A)P(C|A)P(D|B, C)P(E|B, C, D)
CPDs
=C
E
B
A
D
P(A)P(B|A)P(C|A)P(E|B)P(D|C, E)
CPDsGRAPH
![Page 7: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/7.jpg)
Pag.Jan Lemeire / 227When are Causal Models not Good Models?
Meaningful Information of Probability Distributions
meaningful information
Regularities: Conditional Independencies
Kolmogorov Minimal Sufficient Statistic if graph and CPDs are incompressible
Joint Probability Distribution
P(A, B, C, D, E)= C
E
B
A
D
P(A)P(B|A)P(C|A)P(E|B)P(D|C, E)
CPDsGRAPH
![Page 8: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/8.jpg)
Pag.Jan Lemeire / 228When are Causal Models not Good Models?
Representation of Independencies
Graph (DAG) of a Bayesian network is a description of conditional independencies
Faithfulness: All conditional independencies of the distribution are described by the graph.
Theorem: If a faithful Bayesian network exists for a distribution, it is the minimal Bayesian network.
![Page 9: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/9.jpg)
Pag.Jan Lemeire / 229When are Causal Models not Good Models?
Regularities cause unfaithfulness
Theorem: A Bayesian network for which the concatenation of the CPDs is incompressible, is faithful.
C
E
B
A
D
P(A)P(B|A)P(C|A)P(E|B)P(D|C, E)
CPDsGRAPH
C
E
B
A
D
GRAPH
P(A)P(B|A)P(C|A)P(D|B, C)P(E|B, C, D)
CPDs
![Page 10: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/10.jpg)
Pag.Jan Lemeire / 2210When are Causal Models not Good Models?
Overview
The Kolmogorov Minimal Sufficient Statistic (KMSS).
Bayesian Networks as Minimal Descriptions of Probability Distributions.
When the minimal Bayesian network is the KMSS.
When the minimal Bayesian network is NOT the KMSS.
![Page 11: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/11.jpg)
Pag.Jan Lemeire / 2211When are Causal Models not Good Models?
CPDC
CPDE
CPDD
A
B
C
ED
PA
PB
When the minimal Bayesian network is the KMSS.
No other regularities than the independencies present in graph
Model is faithful
KMSS = DAG
quasi-unique and minimal decomposition of the system
CPDs are independent
E
D
CA
B
![Page 12: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/12.jpg)
Pag.Jan Lemeire / 2212When are Causal Models not Good Models?
mechC
mechE
mechD
A
B
C
ED
The Top-Ranked Hypothesis
Each CPD corresponds to an independent part of reality, a mechanism
= Modularity and autonomypossibility to predict the effect of changes to the system (interventions)
Causal component = Reductionism
E
D
CA
B
CPDC
CPDE
CPDD
A
B
C
ED
PA
PB
Comes with certificate
Number 1!!
Highly recommended
![Page 13: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/13.jpg)
Pag.Jan Lemeire / 2213When are Causal Models not Good Models?
Except… World can be More Complex
A C
B
+
+ -
A C
B
True Model
Minimal Model
There is no absolute guarantee, the KMSS might be a bit too simplistic
A C
![Page 14: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/14.jpg)
Pag.Jan Lemeire / 2214When are Causal Models not Good Models?
Great job, Jan!
![Page 15: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/15.jpg)
Pag.Jan Lemeire / 2215When are Causal Models not Good Models?
Overview
The Kolmogorov Minimal Sufficient Statistic (KMSS).
Bayesian Networks as Minimal Descriptions of Probability Distributions.
When the minimal Bayesian network is the KMSS.
When the minimal Bayesian network is NOT the KMSS.
![Page 16: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/16.jpg)
Pag.Jan Lemeire / 2216When are Causal Models not Good Models?
When the minimal Bayesian network is NOT the KMSS…
There are non-modeled regularities, DAG+CPDs is compressible
A. Compressibility of an individual CPDB. Compressibility of a set of CPDs
Case studies:– True Causal Model in set of minimal Bayesian
networks?– Faithfulness?– Modularity?
ALARM
![Page 17: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/17.jpg)
Pag.Jan Lemeire / 2217When are Causal Models not Good Models?
(A1) Local Structure
A single CPD can be described shorter, without affecting the rest of the model– Local structure [Friedman and Goldszmidt, 1996]– Context-specific independencies [Boutilier, 1996]
Everything OK.
A B C P(D|A, B, C)
0 0 0 0.40 0 1 0.60 1 0 0.70 1 1 0.71 0 0 0.31 0 1 0.31 1 0 0.31 1 1 0.3
D
A
P(D| A, B, C)
B C
![Page 18: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/18.jpg)
Pag.Jan Lemeire / 2218When are Causal Models not Good Models?
(A2) Deterministic Relations
Y=f(X)
X YZ XY Z
X ZY
X ZY
Two minimal Bayesian networks.Both unfaithful
X ZY
Violation of the intersection condition
![Page 19: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/19.jpg)
Pag.Jan Lemeire / 2219When are Causal Models not Good Models?
Conclusions for Compressibility of an individual CPD
CPDs are independent modularity is still plausibleTrue Causal Model in set of minimal Bayesian networks!But faithfulness may become invalid– Constraint-based algorithms may fail
![Page 20: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/20.jpg)
Pag.Jan Lemeire / 2220When are Causal Models not Good Models?
(B1) Meta-mechanism
Influence A->B->D and A->C->D exactly balance
UnfaithfulnessLearned correctly We may assume a global mechanism that controls mechanisms such that they neutralize– E.g. evolution
CB
A
D
D A
![Page 21: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/21.jpg)
Pag.Jan Lemeire / 2221When are Causal Models not Good Models?
(B2) Markov Networks
Different model class!!
A
B C
D
A
B C
D
One of the minimal Bayesian Networks.Unfaithful & non-modular
![Page 22: When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008](https://reader035.vdocuments.site/reader035/viewer/2022081518/551ac40d550346856e8b5970/html5/thumbnails/22.jpg)
Pag.Jan Lemeire / 2222When are Causal Models not Good Models?
Conclusions
Faithfulness cannot be guaranteed
Modularity cannot be guaranteed when dependent CPDs
Regularities/Model class under consideration must be properly chosen
Augmentation of Bayesian networks with other qualitative properties
Faithfulness = ability of a model to explicitly explain all regularities of the data