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gR2002gR2002

Peter SpirtesPeter Spirtes

Carnegie Mellon UniversityCarnegie Mellon University

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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

Conditioning

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Manipulating Smoking – First Step

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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