Outline1. MotivationBayes NetworksBayes Networks: UpdatingCausal InferenceCausal Inference2. RepresentationCausation & AssociationDirect CausationCausal GraphsCausal GraphsModeling Ideal InterventionsInterventions & Causal GraphsConditioning vs. InterveningCausal Bayes NetworksStructural Equation ModelsStructural Equation ModelsStructural Equation Models3. Connecting Causation to ProbabilityCausal Markov AxiomCausal Markov ConditionCausal Markov ConditionCausal Markov ConditionCausal Structure Statistical DataCausal Markov AxiomCausal Markov and D-SeparationD-separation: Conditioning vs. Intervening4. Search From Statistical Data to Probability to CausationRepresentations ofD-separation Equivalence ClassesPatterns & PAGsPatternsPatterns: What the Edges MeanPatternsPAGs: Partial Ancestral GraphsPAGs: Partial Ancestral GraphOverview of Search MethodsTetrad 4 Demo5. Regession and Causal InferenceRegression to estimate Causal InfluenceRegression and Causal InferenceLinear RegressionLinear RegressionRegression ExampleRegression ExampleRegression ExampleRegression BiasOngoing ProjectsReferencesGraphical Models --11/30/05 1Causal DiscoveryRichard ScheinesPeter Spirtes, Clark Glymour, and many othersDept. of Philosophy & CALDCarnegie MellonGraphical Models --11/30/05 2Outline1. Motivation2. Representation3. Connecting Causation to Probability (Independence)4. Searching for Causal Models5. Improving on Regression for Causal InferenceGraphical Models --11/30/05 31. MotivationNon-experimental EvidenceTypical Predictive Questions• Can we predict aggressiveness from the amount of violent TV watched• Can we predict crime rates from abortion rates 20 years agoCausal Questions: • Does watching violent TV cause Aggression?• I.e., if we change TV watching, will the level of Aggression change?Day Care AggressivenesJohn Mary A lot None A lot A littleGraphical Models --11/30/05 4Bayes Networks Disease [Heart Disease, Reflux Disease, other] Shortness of Breath[Yes, No] Chest Pain [Yes, No] Qualitative Part:Directed GraphP(Disease = Heart Disease) = .2P(Disease = Reflux Disease) = .5P(Disease = other) = .3P(Chest Pain = yes | D = Heart D.) = .7P(Shortness of B = yes | D= Hear D. ) = .8P(Chest Pain = yes | D = Reflux) = .9P(Shortness of B = yes | D= Reflux ) = .2P(Chest Pain = yes | D = other) = .1P(Shortness of B = yes | D= other ) = .2Quantitative Part:Conditional Probability TablesGraphical Models --11/30/05 5Bayes Networks: UpdatingGiven: Data on SymptomsChest Pain = yesWanted: P(Disease | Chest Pain = yes ) Disease [Heart Disease, Reflux Disease, other] Shortness of Breath[Yes, No] Chest Pain [Yes, No] Updating P(D = Heart Disease) = .2P(D = Reflux Disease) = .5P(D = other) = .3P(Chest Pain = yes | D = Heart D.) = .7P(Shortness of B = yes | D= Hear D. ) = .8P(Chest Pain = yes | D = Reflux) = .9P(Shortness of B = yes | D= Reflux ) = .2P(Chest Pain = yes | D = other) = .1P(Shortness of B = yes | D= other ) = .2Graphical Models --11/30/05 6Causal InferenceGiven: Data on SymptomsChest Pain = yesP(Disease | Chest Pain = yes ) Updating P(Disease | Chest Pain set= yes )Causal InferenceGraphical Models --11/30/05 7Causal InferenceWhen and how can we use non-experimental data to tell us about the effect of an intervention?Manipulated Probability P(Y | X set= x, Z=z)fromUnmanipulated Probability P(Y | X = x, Z=z)Graphical Models --11/30/05 82. Representation1. Association & causal structure - qualitatively2. Interventions3. Statistical Causal Models1. Bayes Networks2. Structural Equation ModelsGraphical Models --11/30/05 9Causation & AssociationX and Y are associated (X _||_ Y) iff∃x1 ≠ x2P(Y | X = x1) ≠ P(Y | X = x2)Association is symmetric: X _||_ Y ⇔ Y _||_ XX is a cause of Y iff∃x1 ≠ x2P(Y | X set= x1) ≠ P(Y | X set= x2)Causation is asymmetric: X Y ⇔ X YGraphical Models --11/30/05 10Direct CausationX is a direct cause of Y relative to S, iff∃z,x1 ≠ x2 P(Y | X set= x1 , Z set= z) ≠ P(Y | X set= x2 , Z set= z)where Z = S -{X,Y}X YGraphical Models --11/30/05 11Causal GraphsCausal Graph G = {V,E} Each edge X → Y represents a direct causal claim:X is a direct cause of Y relative to V ExposureRash Chicken Pox ExposureInfection RashGraphical Models --11/30/05 12Causal GraphsDo Notneed to be Cause CompleteOmitted Causes 2Omitted Causes 1 Exposure Infection Symptoms Doneed to be Common Cause Complete Exposure Infection Symptoms Omitted Common CausesGraphical Models --11/30/05 13Modeling Ideal InterventionsIdeal Interventions (on a variable X):• Completely determinethe value or distribution of a variable X• Directly Target only X (no “fat hand”)E.g., Variables: Confidence, Athletic PerformanceIntervention 1: hypnosis for confidenceIntervention 2: anti-anxiety drug (also muscle relaxer)Graphical Models --11/30/05 14Modeling Ideal InterventionsInterventions on the EffectPre-experimental SystemPostSweaters OnRoom TemperatureGraphical Models --11/30/05 15Modeling Ideal InterventionsInterventions on the CausePre-experimental SystemPostSweatersOnRoom TemperatureGraphical Models --11/30/05 16Interventions & Causal Graphs• Model an ideal intervention by adding an “intervention” variable outside the original system• Erase all arrows pointing into the variable intervened upon Intervene to change InfPost-intervention graph?Pre-intervention graphExpInf RashI Exp InfRashGraphical Models --11/30/05 17Conditioning vs. InterveningP(Y | X = x1) vs. P(Y | X set= x1)Teeth SlidesGraphical Models --11/30/05 18Causal Bayes Networks Smoking [0,1] Lung Cancer[0,1]Yellow Fingers[0,1]The Joint Distribution FactorsAccording to the Causal Graph,i.e., for all X in VP(V) = ΠP(X|Immediate Causes of(X))P(S = 0) = .7P(S = 1) = .3P(YF = 0 | S = 0) = .99 P(LC = 0 | S = 0) = .95P(YF = 1 | S = 0) = .01 P(LC = 1 | S = 0) = .05P(YF = 0 | S = 1) = .20 P(LC = 0 | S = 1) = .80P(YF = 1 | S = 1) = .80 P(LC = 1 | S = 1) = .20P(S,YF, L) = P(S) P(YF | S) P(LC | S)Graphical Models --11/30/05 19Structural Equation Models Education LongevityIncomeCausal GraphStatistical Model1. Structural Equations2. Statistical ConstraintsGraphical Models --11/30/05 20Structural Equation Models Education LongevityIncomeCausal Graphz Structural Equations:One Equation for each variable V in the graph:V = f(parents(V), errorV)for SEM (linear regression) f is a linear functionz Statistical Constraints:Joint Distribution over the Error termsGraphical Models --11/30/05 21Structural Equation
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