Causal Performance Modelsparallel.vub.ac.be/~jan/papers/JanLemeire_WhenAreCausal...Jan Lemeire Pag....
Transcript of Causal Performance Modelsparallel.vub.ac.be/~jan/papers/JanLemeire_WhenAreCausal...Jan Lemeire Pag....
When are Graphical Causal Models not Good Models?
CAPITS 2008
Jan Lemeire
September 12th 2008
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.
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.
Pag.Jan Lemeire / 224When are Causal Models not Good Models?
Randomness versus Regularity
001001001001001001001001001001001001001
011000110101101010111001001101000101110
Random string: incompressible, maximal complexity
But: it is no meaningful information, only accidentalinformation
Regular string: compressible, low complexity
repetition 12 times, 001regularities randomness
Model that minimally describes regularities (qualitative props)
= Kolmogorov Minimal Sufficient Statistic (KMSS)
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.
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
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
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.
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
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.
Pag.Jan Lemeire / 2211When are Causal Models not Good Models?
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
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 autonomy
possibility to predict the effect of changes to the system (interventions)
Causal component = Reductionism
Comes with
certificate
Num
ber
1!!
Highly
recommended
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
Pag.Jan Lemeire / 2214When are Causal Models not Good Models?
Great job, Jan!
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.
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 CPD
B. Compressibility of a set of CPDs
Case studies:– True Causal Model in set of minimal Bayesian
networks?
– Faithfulness?
– Modularity?
ALARM
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 0.4
0 0 1 0.6
0 1 0 0.7
0 1 1 0.7
1 0 0 0.3
1 0 1 0.3
1 1 0 0.3
1 1 1 0.3
D
A
P(D| A, B, C)
B C
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
Pag.Jan Lemeire / 2219When are Causal Models not Good Models?
Conclusions for Compressibility of an individual CPD
CPDs are independent
modularity is still plausible
True Causal Model in set of minimal Bayesian networks!
But faithfulness may become invalid
– Constraint-based algorithms may fail
Pag.Jan Lemeire / 2220When are Causal Models not Good Models?
(B1) Meta-mechanism
Influence A->B->D and
A->C->D exactly balance
Unfaithfulness
Learned correctly
We may assume a global mechanism that controls mechanisms such that they neutralize
– E.g. evolution
CB
A
D
D A
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
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