1Graphical Models --11/29/06 1Causal DiscoveryRichard ScheinesPeter Spirtes, Clark Glymour, and many othersDept. of Philosophy & Machine LearningCarnegie MellonGraphical Models --11/29/06 2Outline1. Motivation2. Representation3. Connecting Causation to Probability (Independence)4. Searching for Causal Models2Graphical Models --11/29/06 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 Aggressiveness John Mary A lot None A lot A little Graphical Models --11/29/06 4Bayes Networks Disease [Heart Disease, Reflux Disease, other] Shortness of Breath [Yes, No] Chest Pain [Yes, No] Qualitative Part:Directed GraphQuantitative Part:Conditional Probability TablesP(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 ) = .23Graphical Models --11/29/06 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 --11/29/06 6Causal InferenceGiven: Data on SymptomsChest Pain = yesP(Disease | Chest Pain = yes ) Updating P(Disease | Chest Pain set= yes )Causal Inference4Graphical Models --11/29/06 7Causal InferenceManipulated Probability P(Y | X set= x, Z=z)fromUnmanipulated Probability P(Y | X = x, Z=z) When and how can we use non-experimental data to tell us about the effect of an intervention?Graphical Models --11/29/06 8Conditioning ≠ InterveningP(Y | X = x1) vs. P(Y | X set= x1)Teeth Slides5Graphical Models --11/29/06 92. Representation1. Association & causal structure - qualitatively2. Interventions3. Statistical Causal Models1. Bayes Networks2. Structural Equation ModelsGraphical Models --11/29/06 10Causation & AssociationX is a cause of Y iff∃x1 ≠ x2P(Y | X set= x1) ≠ P(Y | X set= x2)Causation is asymmetric: X Y ⇔ X YX and Y are associated (X _||_ Y) iff∃x1 ≠ x2P(Y | X = x1) ≠ P(Y | X = x2)Association is symmetric: X _||_ Y ⇔ Y _||_ X6Graphical Models --11/29/06 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 Exposure Rash Exposure Infection Rash Chicken PoxGraphical Models --11/29/06 12Causal GraphsDo Notneed to be Cause CompleteDoneed to be Common Cause Complete Exposure Infection Symptoms Exposure Infection Symptoms Omitted Common Causes Omitted Causes 2Omitted Causes 17Graphical Models --11/29/06 13Modeling Ideal InterventionsIdeal Interventions (on a variable X):(on a variable X):(on a variable X):(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 --11/29/06 14Sweaters OnRoom TemperaturePre-experimental SystemPostModeling Ideal InterventionsInterventions on the Effect8Graphical Models --11/29/06 15Modeling Ideal InterventionsSweatersOnRoom TemperaturePre-experimental SystemPostInterventions on the CauseGraphical Models --11/29/06 16Interventions & Causal GraphsModel an ideal intervention by adding an “intervention” variable outside the original system as a direct cause of its target. Education Income Taxes Pre-intervention graphIntervene on Income“Soft” Intervention Education Income Taxes I “Hard” Intervention Education Income Taxes I9Graphical Models --11/29/06 17Causal Bayes NetworksP(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) = .20 S m o k in g [ 0 ,1 ] L u n g C a n c e r[0 ,1 ]Y e llo w F in g e rs[ 0 ,1 ]P(S,YF, L) = P(S) P(YF | S) P(LC | S)The Joint Distribution FactorsAccording to the Causal Graph,i.e., for all X in VP(V) = ΠP(X|Immediate Causes of(X))Graphical Models --11/29/06 18Structural Equation Models1. Structural Equations2. Statistical Constraints Education LongevityIncomeStatistical ModelCausal Graph10Graphical Models --11/29/06 19Structural Equation Modelsz 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 terms Education LongevityIncomeCausal GraphGraphical Models --11/29/06 20Structural Equation ModelsEquations:Education = εedIncome = β1 Education + εincomeLongevity = β2 Education + εLongevityStatistical Constraints:(εed, εIncome,εIncome ) ~N(0,Σ2) − Σ2 diagonal- no variance is zero Education LongevityIncomeCausal Graph Education εIncome εLongevity β1 β2 LongevityIncomeSEM Graph(path diagram)11Graphical Models --11/29/06 213. ConnectingCausation to ProbabilityGraphical Models --11/29/06 22Semantics of Causation Choice 1: Define X Y, or X Y in terms of intervention, i.e., (hypothetical) treatment) Choice 2: Causal systems over V ⇒Probabilistic Independence Relations in P(V)12Graphical Models --11/29/06 23X is a cause of Y iff∃x1 ≠ x2P(Y | X set= x1) ≠ P(Y | X set= x2)Causation is asymmetric: X Y ⇔ X YX and Y are associated (X _||_ Y) iff∃x1 ≠ x2P(Y | X = x1) ≠ P(Y | X = x2)Association is symmetric: X _||_ Y ⇔ Y _||_ XChoice 1: Define Causation from ManipulationGraphical Models --11/29/06 24X 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 Y Choice 1: Define Direct Causation from Intervention13Graphical Models --11/29/06
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