1 Tetrad: Machine Learning and Graphcial Causal Models Richard Scheines Joe Ramsey Carnegie Mellon University Peter Spirtes, Clark Glymour

Goals 1)Convey rudiments of graphical causal models 2)Basic working knowledge of Tetrad IV 2

Tetrad IV: Complete Causal Modeling Tool 3

Tetrad 1)Main website: 2)Download site: 3)Data files: 4

Topic Outline 1)Motivation 2)Representing/Modeling Causal Systems 3)Estimation and Updating 4)Model Search 5)Linear Latent Variable Models 6)Case Study: fMRI 5

Statistical Causal Models: Goals 1)Policy, Law, and Science: How can we use data to answer a)subjunctive questions (effects of future policy interventions), or b)counterfactual questions (what would have happened had things been done differently (law)? c)scientific questions (what mechanisms run the world) 2)Rumsfeld Problem: Do we know what we do and don’t know: Can we tell when there is or is not enough information in the data to answer causal questions? 6

Causal Inference Requires More than Probability In general: P(Y=y | X=x, Z=z) ≠ P(Y=y | X set =x, Z=z) Prediction from Observation ≠ Prediction from Intervention P(Lung Cancer 1960 = y | Tar-stained fingers 1950 = no) Causal Prediction vs. Statistical Prediction: Non-experimental data (observational study) Background Knowledge P(Y,X,Z) P(Y=y | X=x, Z=z) Causal Structure P(Y=y | X set =x, Z=z) ≠ P(Lung Cancer 1960 = y | Tar-stained fingers 1950 set = no) 7

Foundations of Causal Epistemology Some Causal Structures can parameterize the same set of probability distributions, some cannot 8 X ZY X ZY X ZY X ZY P 2 (X,YZ) P 1 (X,YZ)

Causal Search 9 Causal Search: 1.Find/compute all the causal models that are indistinguishable given background knowledge and data 2.Represent features common to all such models Multiple Regression is often the wrong tool for Causal Search: Example: Foreign Investment & Democracy

10 Foreign Investment Does Foreign Investment in 3 rd World Countries inhibit Democracy? Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, N = 72 POdegree of political exclusivity CVlack of civil liberties ENenergy consumption per capita (economic development) FIlevel of foreign investment

11 Correlations po fi en fi en cv Foreign Investment

12 Regression Results po =.227*fi -.176*en +.880*cv SE (.058) (.059) (.060) t Interpretation: foreign investment increases political repression Case Study 1: Foreign Investment

Alternatives Case Study 1: Foreign Investment There is no model with testable constraints (df > 0) in which FI has a positive effect on PO that is not rejected by the data.

Outline 1)Motivation 2)Representing/Modeling Causal Systems 1)Causal Graphs 2)Standard Parametric Models 1)Bayes Nets 2)Structural Equation Models 3)Other Parametric Models 1)Generalized SEMs 2)Time Lag models 14

15 Causal Graph G = {V,E} Each edge X  Y represents a direct causal claim: X is a direct cause of Y relative to V Causal Graphs Years of Education Income Skills and Knowledge Years of Education

16 Causal Graphs Not Cause Complete Common Cause Complete

17 Sweaters On Room Temperature Pre-experimental SystemPost Modeling Ideal Interventions Interventions on the Effect

18 Modeling Ideal Interventions Sweaters On Room Temperature Pre-experimental SystemPost Interventions on the Cause

19 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 “Soft” Intervention “Hard” Intervention

20 Tetrad Demo Build and Save an acyclic causal graph: 1)with 3 measured variables, no latents 2)with at least 3 measured variables, and at least 1 latent

21 Parametric Models

22 Causal Bayes Networks P(S,YF, L) = P(S) P(YF | S) P(LC | S) The Joint Distribution Factors According to the Causal Graph,

23 Causal Bayes Networks P(S = 0) =  1 P(S = 1) = 1 -  1 P(YF = 0 | S = 0) =  2 P(LC = 0 | S = 0) =  4 P(YF = 1 | S = 0) = 1-  2 P(LC = 1 | S = 0) = 1-  4 P(YF = 0 | S = 1) =  3 P(LC = 0 | S = 1) =  5 P(YF = 1 | S = 1) = 1-  3 P(LC = 1 | S = 1) = 1-  5 P(S) P(YF | S) P(LC | S) = f(  ) The Joint Distribution Factors According to the Causal Graph, All variables binary [0,1]:  = {  1,  2,  3,  4,  5, }

24 Tetrad Demo

25 Structural Equation Models zStructural Equations F or each variable X  V, an assignment equation: X := f X (immediate-causes(X),  X ) Causal Graph zExogenous Distribution : Joint distribution over the exogenous vars : P(  )

26 Equations: Education :=  Education Income :=    Education  income Longevity :=    Education  Longevity Causal Graph Path diagram Linear Structural Equation Models E.g. (  ed,  Income,  Income ) ~N(0,  2 )  2 diagonal, - no variance is zero Exogenous Distribution: P(  ed,  Income,  Income ) -  i≠j  i   j (pairwise independence) - no variance is zero Structural Equation Model: V = BV + E

27 Tetrad Demo 1)Interpret your causal graph with 3 measured variables with at least 2 parametric models: a)Bayes Parametric Model b)SEM Parametric Model 2)Interpret your other graph with a parametric model of your choice

28 Instantiated Models

29 Tetrad Demo 1)Instantiate at least one Bayes PM with a Bayes IM 2)Instantiate at least one SEM PM with a SEM IM 3)Instantiate at least one SEM PM with a Standardized SEM IM 4)Generate two data sets (N= 50, N=5,000) for each

Outline 1)Motivation 2)Representing/Modeling Causal Systems 1)Causal Graphs 2)Standard Parametric Models 1)Bayes Nets 2)Structural Equation Models 3)Other Parametric Models 1)Generalized SEMs 2)Time Lag models 30

Generalized SEM 1)The Generalized SEM is a generalization of the linear SEM model. 2)Allows for arbitrary connection functions 3)Allows for arbitrary distributions 4)Simulation from cyclic models supported.

Hands On 1)Create a DAG. 2)Parameterize it as a Generalized SEM. 3)Open the Generalized SEM and select Apply Templates from the Tools menu. 4)Apply the default template to variables, which will make them all linear functions. 5)For errors, select a non-Gaussian distribution, such as U(0, 1). 6)Save.

Time Series Simulation (Hands On) 1)Tetrad includes support for doing time series simulations. 2)First, one creates a time series graph. 3)Then one parameterizes the time series graph as a SEM. 4)Then one instantiates the SEM. 5)Then one simulates data from the SEM Instantiated Model.

Time Series Simulation One can, e.g., calculate a vector auto-regression for it. (One can do this as well from time series data loaded in.) Attach a data manipulation box to the data. Select vector auto-regression. One can create staggered time series data Attach a data manipulation box. Select create time series data. Should give the time lag graph with some extra edges in the highest lag.

35 Estimation

36 Tetrad Demo 1)Estimate one Bayes PM for which you have an IM and data 2)Estimate one SEM PM for which you have an IM and data 3)Import data from charity.txt, and build and estimate model two models to estimate on those data

Hypothesis 1 37 Hypothesis 2

38 Updating

39 Tetrad Demo 1)Pick one of your Bayes IMs 2)Find a variable X to update conditional on Y such that: The marginal on X changes when Y is passively observed = y, but does not change when Y is manipulated = y 3)Find a variable Z to update conditional on W such that: The marginal on Z changes when W is passively observed = w, and changes in exactly the same way when W is manipulated = w