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1 Extra Slides

2 Causal Graphs Causal Graph G = {V,E}
Each edge X  Y represents a direct causal claim: X is a direct cause of Y relative to V Years of Education Income 1. don’t define causality - but will introduce axioms to connect probability to causality 2. many fields proceed without agreement on definition - probability, “force” in mechanics, interpretation of quantum mechanics, etc. 3. a number of different kinds of graphs represent probability distributions and independence - advantage of directed graphs is also represents causal relations 4. will introduce several extensions Income Skills and Knowledge Years of Education

3 Omitteed Common Causes
Causal Graphs Omitteed Causes Not Cause Complete Income Skills and Knowledge Years of Education Common Cause Complete Omitteed Common Causes 1. don’t define causality - but will introduce axioms to connect probability to causality 2. many fields proceed without agreement on definition - probability, “force” in mechanics, interpretation of quantum mechanics, etc. 3. a number of different kinds of graphs represent probability distributions and independence - advantage of directed graphs is also represents causal relations 4. will introduce several extensions Income Skills and Knowledge Years of Education

4 Modeling Ideal Interventions
Interventions on the Effect Post Pre-experimental System Room Temperature Sweaters On

5 Modeling Ideal Interventions
Interventions on the Cause Post Pre-experimental System Sweaters On Room Temperature

6 Interventions & Causal Graphs
Model an ideal intervention by adding an “intervention” variable outside the original system as a direct cause of its target. Pre-intervention graph Intervene on Income “Hard” Intervention Fat Hand - intervention - cholesterol drug -- arythmia “Soft” Intervention

7 Interventions & Causal Graphs
X5 X2 X1 Pre-intervention Graph X4 X3 X6 Intervention: hard intervention on both X1, X4 Soft intervention on X3 X1 X2 X3 X4 X6 X5 I S Fat Hand - intervention - cholesterol drug -- arythmia Post-Intervention Graph?

8 Interventions & Causal Graphs
X5 X2 X1 Pre-intervention Graph X4 X3 X6 Intervention: hard intervention on both X1, X4 Soft intervention on X3 X1 X2 X3 X4 X6 X5 I S Fat Hand - intervention - cholesterol drug -- arythmia Post-Intervention Graph?

9 Interventions & Causal Graphs
X5 X2 X1 Pre-intervention Graph X4 X3 X6 Intervention: hard intervention on X3 Soft interventions on X6, X4 I S X1 X2 X3 X4 X6 X5 Fat Hand - intervention - cholesterol drug -- arythmia Post-Intervention Graph?

10 Interventions & Causal Graphs
Smoking Pre-intervention Graph Stained_Teeth LC Trek between Stained_Teeth and LC In Pre-Intervention Graph? Treks  Association Yes  Stained_Teeth _||_ LC Smoking Paint Teeth White Stained_Teeth LC Fat Hand - intervention - cholesterol drug -- arythmia Trek between Stained_Teeth and LC In Post-Intervention Graph? Treks  Association No  Stained_Teeth _||_m LC

11 Calculating the effect of a hard interventions
P(YF,S,L) = P(S) P(YF|S) P(L|S) Pm (YF,S,L) = P(S) P(L|S) P(YF| I)

12 Calculating the effect of a hard intervention
P(S,YF, L) = P(S) P(YF | S) P(LC | S) P(S=1,YF=1, LC=1) = * * = .048 P(YF =1 | I ) = .5 Pm (S=1,YFset=1, LC=1) = ? Pm (S=1,YFset=1, LC=1) = P(S) P(YF | I) P(LC | S) Pm (S=1,YFset=1, LC=1) = .3 * * = .03

13 Calculating the effect of a soft intervention
P(YF,S,L) = P(S) P(YF|S) P(L|S) Pm (YF,S,L) = P(S) P(L|S) P(YF| S, Soft)

14 Independence Equivalence Classes: Patterns & PAGs
Patterns (Verma and Pearl, 1990): graphical representation of d-separation equivalence class (among models with no latent common causes) PAGs: (Richardson 1994) graphical representation of a d-separation equivalence class that includes models with latent common causes and sample selection bias that are d-separation equivalent over a set of measured variables X

15 Patterns 1. represents set of conditional independence and distribution equivalent graphs 2. same adjacencies 3. undirected edges mean some contain edge one way, some contain other way 4. directed edge means they all go same way 5. Pearl and Verma -complete rules for generating from Meek, Andersson, Perlman, and Madigan, and Chickering 6. instance of chain graph 7. since data can’t distinguish, in absence of background knowledge is right output for search 8. what are they good for?

16 Patterns: What the Edges Mean
1. represents set of conditional independence and distribution equivalent graphs 2. same adjacencies 3. undirected edges mean some contain edge one way, some contain other way 4. directed edge means they all go same way 5. Pearl and Verma -complete rules for generating from Meek, Andersson, Perlman, and Madigan, and Chickering 6. instance of chain graph 7. since data can’t distinguish, in absence of background knowledge is right output for search 8. what are they good for?

17 Patterns 1. represents set of conditional independence and distribution equivalent graphs 2. same adjacencies 3. undirected edges mean some contain edge one way, some contain other way 4. directed edge means they all go same way 5. Pearl and Verma -complete rules for generating from Meek, Andersson, Perlman, and Madigan, and Chickering 6. instance of chain graph 7. since data can’t distinguish, in absence of background knowledge is right output for search 8. what are they good for?

18 Patterns Specify all the causal graphs represented by the Pattern:
Why not? 1. represents set of conditional independence and distribution equivalent graphs 2. same adjacencies 3. undirected edges mean some contain edge one way, some contain other way 4. directed edge means they all go same way 5. Pearl and Verma -complete rules for generating from Meek, Andersson, Perlman, and Madigan, and Chickering 6. instance of chain graph 7. since data can’t distinguish, in absence of background knowledge is right output for search 8. what are they good for?

19 Causal Search Spaces are Large
Directed Acyclic Graphs (between 2 𝑁 2 and 3 𝑁 2 ) … 𝑁 2 is O(N2) Directed Graphs ( 4 𝑁 2 ) Markov Equivalence Class of DAGs (patterns) : DAGs / 3.7 Markov Equivalence Class of DAGs with confounders (roughly PAGs) ?? Equivalence Class of “Linear Measurement Models” ?? Equivalence Class of Directed Graphs with confounders Relative to: Experimental Setup V = {Obs, Manip} ?? 1

20 Causal Search as a Method
Causal Knowledge e.g., Markov Equivalence Class of Causal Graphs Experimental Setup(V) V = {Obs, Manip} P(Manip) Statistical Inference Discovery Algorithm PManip(V) Background Knowledge Salary  Gender Infection  Symptoms General Assumptions Markov, Faithfulness Linearity Gaussianity Acyclicity Data 1

21 For Example Passive Observation Statistical Inference
Background Knowledge X2 prior in time to X3 General Assumptions Markov, Faithfulness, No latents, no cycles,

22 Faithfulness Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G. Tax Rate b3 Revenues := b1Rate + b2Economy + eRev Economy := b3Rate + eEcon b1 Economy b2 Tax Revenues Faithfulness: b1 ≠ -b3b2 b2 ≠ -b3b1 1

23 Faithfulness Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G. All and only the constraints that hold in P(V) are entailed by the causal structure G(V), rather than lower dimensional surfaces in the parameter space. E.g., Weak Causal Markov Axiom: X and Y causally disconnected ╞ X _||_ Y Faithfulness: X and Y causally disconnected ╡ X _||_ Y 1

24 Challenges to Faithfulness
Gene A - By evolutionary design: Gene A _||_ Protein 24 Gene B + + Protein 24 By evolutionary design: Air temp _||_ Core Body Temp Air Temp Core Body Temp Homeostatic Regulator Sampling Rate vs. Equilibration rate 1

25 A Few Causal Discovery Highlights

26 Autism Catherine Hanson, Rutgers ASD vs. NT Usual Approach:
Search for differential recruitment of brain regions

27 ASD vs. NT Causal Modeling Approach: Examine connectivity of ROIs
Face processing network Theory of Mind network Action understanding network

28 Results FACE TOM ACTION

29 face processing: ASD  NT
What was Learned face processing: ASD  NT Theory of Mind: ASD ≠ NT action understanding: ASD ≠ NT when faces involved

30 Other Applications Educational Research: Economics: Lead and IQ
Online Courses, MOOCs (the “Doer” effect) Cog. Tutors Economics: Causes of Meat Prices, Effects of International Trade Lead and IQ Stress, Depression, Religiosity Climate Change Modeling The Effects of Welfare Reform Etc. ! 1

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