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Graphs often given causal interpretationGraphs often given causal interpretation Graphs can be used to represent both causal Graphs can be used to represent both causal
hypotheses and probability distributionshypotheses and probability distributions e.g. in a directed acyclic graph (DAG) A e.g. in a directed acyclic graph (DAG) A B means A B means A
is a direct cause of Bis a direct cause of B DAG also represents a set of distributions sharing DAG also represents a set of distributions sharing
conditional independence relationsconditional independence relations Causal interpretation is common in social science Causal interpretation is common in social science
applications (structural equation modelling)applications (structural equation modelling) Causal representation of genetic regulatory Causal representation of genetic regulatory
networksnetworks
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TETRADTETRAD
Dedicated to search for causal models under a variety of Dedicated to search for causal models under a variety of different assumptions about what is knowndifferent assumptions about what is known Has several different kinds of graphs, depending upon background Has several different kinds of graphs, depending upon background
assumptionsassumptions Has a number of different kinds of search strategiesHas a number of different kinds of search strategies Allows some explicit representation of background knowledgeAllows some explicit representation of background knowledge Has some modules for calculating equivalence class of given graphHas some modules for calculating equivalence class of given graph Recently developed graphical interfaceRecently developed graphical interface Should have module for calculating effects of interventionsShould have module for calculating effects of interventions
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The causal interpretation of graphical models suggests several unusual operations
The causal interpretation of graphical models suggests several unusual operations Calculation of effect of manipulationCalculation of effect of manipulation Calculation of equivalence class (aid to Calculation of equivalence class (aid to
calculation of effect of manipulation)calculation of effect of manipulation)
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Kinds of graphical models in TETRADKinds of graphical models in TETRAD
Directed acyclic graphs (discrete, normal)Directed acyclic graphs (discrete, normal) Directed cyclic graphs (normal)Directed cyclic graphs (normal) Pattern Pattern Mixed ancestral graphs (normal)Mixed ancestral graphs (normal) Partial ancestral graphsPartial ancestral graphs
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Difference between calculation of manipulation versus conditioningDifference between calculation of manipulation versus conditioning In conditioning, the result depends only In conditioning, the result depends only
upon the joint distribution and the event upon the joint distribution and the event conditioned on, conditioned on,
In manipulating, the results depend upon the In manipulating, the results depend upon the joint distribution, the event manipulated, joint distribution, the event manipulated, and the causal relations among the variablesand the causal relations among the variables This means that locating alternative good This means that locating alternative good
models is essential for correct prediction of models is essential for correct prediction of manipulationmanipulation
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P(Lung Cancer = yes||Smoking = yes) = ¾ =
Manipulating Smoking – After waiting
P(Lung Cancer = yes|Smoking = yes) = ¾
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Calculation of effect of manipulationCalculation of effect of manipulation When there are no latent variables and When there are no latent variables and
structure is known - simplestructure is known - simple When there are latent variables and the When there are latent variables and the
structure is known (Pearl 2001)structure is known (Pearl 2001) When the structure is partially known (SGS When the structure is partially known (SGS
2001)2001)
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Calculation of Effect of Manipulation – Equivalence ClassCalculation of Effect of Manipulation – Equivalence ClassA A
B CB C
D D
GG11
A A
B CB C
D D
GG22
A A
B CB C
D D
Pattern Pattern
G1 and G2 represent the same distribution, agree on the effect on D of manipulating B, disagree about the effect on A of manipulating B
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Calculation of Effect of Manipulation – Equivalence ClassesCalculation of Effect of Manipulation – Equivalence Classes
A A
B CB C
D D
GG11
A A
B CB C
D D
GG22
A A
B CB C
D D
Pattern Pattern
Pattern represents the equivalence class of DAGs if there are no latent variables. PAG represents the equivalence class of DAGs if there might be latent variables.
A A
B CB C
D D
PAG PAG
o
o o
o
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Edge types in different graphsEdge types in different graphs
oooo oooo combinations of edges subject to varying combinations of edges subject to varying
constraintsconstraints
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The Statistical Theory for some graphical models is only partially worked out
The Statistical Theory for some graphical models is only partially worked out
MAGs and PAGsMAGs and PAGs know how to parameterize in linear casesknow how to parameterize in linear cases may not be a unique maximum likelihood may not be a unique maximum likelihood
estimateestimate PAG – not known how to efficiently determine PAG – not known how to efficiently determine
if arbitrary combination of edges is PAGif arbitrary combination of edges is PAG
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Specific searchesSpecific searches
Assuming no latent variables or cyclesAssuming no latent variables or cycles Hill climbing – BIC, posterior probability Hill climbing – BIC, posterior probability
(normal, discrete)(normal, discrete) Constraint based – PC (normal, discrete)Constraint based – PC (normal, discrete) CombinedCombined
Assuming no cyclesAssuming no cycles Hill climbing – BIC (normal)Hill climbing – BIC (normal) Constraint based – FCI (normal, discrete)Constraint based – FCI (normal, discrete)
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Other featuresOther features
Estimate parameters - DAGs (discrete, normal)Estimate parameters - DAGs (discrete, normal) Representation of background knowledgeRepresentation of background knowledge Find equivalence class of given DAG (no latents, Find equivalence class of given DAG (no latents,
possibly cyclic)possibly cyclic) Graphical interfaceGraphical interface Should have module to calculate effects of Should have module to calculate effects of
manipulationsmanipulations Known structure, no latentsKnown structure, no latents Known structure, latent variablesKnown structure, latent variables Partially known structurePartially known structure
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As a probabilistic model graphical models require usual operationsAs a probabilistic model graphical models require usual operations As a probabilistic model, it requires the As a probabilistic model, it requires the
usual set of proceduresusual set of procedures SearchSearch EstimationEstimation TestingTesting ScoringScoring ConditioningConditioning
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SummarySummary
The causal interpretation of graphical The causal interpretation of graphical models offers an opportunity to provide models offers an opportunity to provide functionality not found in most other kinds functionality not found in most other kinds of models (e.g. predicting affects of of models (e.g. predicting affects of manipulations)manipulations)
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SummarySummary
Added functionality, different domains and Added functionality, different domains and different background knowledge require a different background knowledge require a variety of different kinds of graphical variety of different kinds of graphical modelsmodels desirability of flexibility in graphical desirability of flexibility in graphical
representationrepresentation desirability of allowing each type to inherit as desirability of allowing each type to inherit as
much as possible from more general much as possible from more general representationsrepresentations
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SummarySummary
Because of the need to locate good Because of the need to locate good alternative modelsalternative models Search plays a very important role (score-based, Search plays a very important role (score-based,
constraint-based, and combinations)constraint-based, and combinations) Calculating equivalence classes is essentialCalculating equivalence classes is essential Collection and representation of background Collection and representation of background
knowledge to guide search is very importantknowledge to guide search is very important
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