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When are Graphical Causal Models not Good Models? CAPITS 2008 Jan Lemeire September 12 th 2008

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Pag. Jan Lemeire / 22 2 When are Causal Models not Good Models? 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.

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Pag. Jan Lemeire / 22 3 When are Causal Models not Good Models? 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.

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Pag. Jan Lemeire / 22 4 When are Causal Models not Good Models? Randomness versus Regularity 001001001001001001001001001001001001001 011000110101101010111001001101000101110 Random string: incompressible, maximal complexity But: it is no meaningful information, only accidental information Regular string: compressible, low complexity repetition 12 times, 001 regularities randomness Model that minimally describes regularities (qualitative props) = Kolmogorov Minimal Sufficient Statistic (KMSS)

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Pag. Jan Lemeire / 22 5 When are Causal Models not Good Models? 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.

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Pag. Jan Lemeire / 22 6 When are Causal Models not Good Models? Description of Probability Distributions with Bayesian networks Multiple Bayesian networks select minimal one

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Pag. Jan Lemeire / 22 7 When 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

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Pag. Jan Lemeire / 22 8 When 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.

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Pag. Jan Lemeire / 22 9 When are Causal Models not Good Models? Regularities cause unfaithfulness Theorem: A Bayesian network for which the concatenation of the CPDs is incompressible, is faithful.

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Pag. Jan Lemeire / 22 10 When are Causal Models not Good Models? 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.

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Pag. Jan Lemeire / 22 11 When 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

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Pag. Jan Lemeire / 22 12 When are Causal Models not Good Models? 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

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Pag. Jan Lemeire / 22 13 When are Causal Models not Good Models? Except… World can be More Complex True Model Minimal Model There is no absolute guarantee, the KMSS might be a bit too simplistic

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Pag. Jan Lemeire / 22 14 When are Causal Models not Good Models? Great job, Jan!

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Pag. Jan Lemeire / 22 15 When are Causal Models not Good Models? 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.

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Pag. Jan Lemeire / 22 16 When 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?

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Pag. Jan Lemeire / 22 17 When 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. ABCP(D|A, B, C) 0000.4 0010.6 0100.7 011 1000.3 101 110 111

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Pag. Jan Lemeire / 22 18 When are Causal Models not Good Models? (A2) Deterministic Relations Two minimal Bayesian networks. Both unfaithful Violation of the intersection condition

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Pag. Jan Lemeire / 22 19 When 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

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Pag. Jan Lemeire / 22 20 When 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

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Pag. Jan Lemeire / 22 21 When are Causal Models not Good Models? (B2) Markov Networks Different model class!! One of the minimal Bayesian Networks. Unfaithful & non-modular

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Pag. Jan Lemeire / 22 22 When 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

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