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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

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Tetrad IV: Complete Causal Modeling Tool

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Tetrad

1) Main website: http://www.phil.cmu.edu/projects/tetrad/

2) Download site: http://www.phil.cmu.edu/projects/tetrad_download/

3) Data files:

www.phil.cmu.edu/projects/tetrad_download/download/workshop/Data

/

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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

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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?

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Causal Inference Requires More than Probability

In general: P(Y=y | X=x, Z=z) ≠ P(Y=y | Xset=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 | Xset=x, Z=z)

≠ P(Lung Cancer 1960 = y | Tar-stained fingers 1950 set = no)

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Foundations of Causal Epistemology

Some Causal Structures can parameterize the

same set of probability distributions, some cannot

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X ZY

X ZY

X ZY

X ZY

P2(X,YZ)

P1(X,YZ)

Causal Search

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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

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Foreign Investment

Does Foreign Investment in 3rd World Countries inhibit Democracy?

Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, 141-146.

N = 72

PO degree of political exclusivity

CV lack of civil liberties

EN energy consumption per capita (economic development)

FI level of foreign investment

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Correlations

po fi en fi -.175 en -.480 0.330 cv 0.868 -.391 -.430

Foreign Investment

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Regression Results

po = .227*fi - .176*en + .880*cv

SE (.058) (.059) (.060)

t 3.941 -2.99 14.6

Interpretation: foreign investment increases political repression

Case Study 1: Foreign Investment

Alternatives

.217

FI

PO

CV En

Regression

.88 -.176

FI

PO

CV En

Tetrad - FCI

FI

PO

CV En

Fit: df=2, 2=0.12, p-value = .94

.31 -.23

.86 -.48

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

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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

IncomeSkills and Knowledge

Years of Education

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Causal Graphs

Not Cause Complete

Common Cause Complete

Education Income Happiness

Omitted Causes

Omitted

Common Causes

Education Income Happiness

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Sweaters On

Room Temperature

Pre-experimental SystemPost

Modeling Ideal Interventions

Interventions on the Effect

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Modeling Ideal Interventions

SweatersOn

Room Temperature

Pre-experimental SystemPost

Interventions on the Cause

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Interventions & 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 graph

Intervene on Income

“Soft” Intervention

Education Income Taxes

I

“Hard” Intervention

Education Income Taxes

I

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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

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Parametric Models

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Causal Bayes Networks

Smoking [0,1]

Lung Cancer[0,1]

Yellow Fingers[0,1]

P(S,YF, L) = P(S) P(YF | S) P(LC | S)

The Joint Distribution Factors

According to the Causal Graph,

))(_|()(

Vx

XcausesDirectXVP P

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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

Smoking [0,1]

Lung Cancer[0,1]

Yellow Fingers[0,1]

P(S) P(YF | S) P(LC | S) = f()

The Joint Distribution Factors

According to the Causal Graph,

))(_|()(

Vx

XcausesDirectXVP P

All variables binary [0,1]: = {1, 2,3,4,5, }

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Tetrad Demo

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Structural Equation Models

Structural EquationsFor each variable X V, an assignment equation:

X := fX(immediate-causes(X), eX)

Education

LongevityIncome

Causal Graph

Exogenous Distribution: Joint distribution over the exogenous vars : P(e)

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Equations: Education := Education

Income :=Educationincome

Longevity :=EducationLongevit

y

Education

LongevityIncome

Causal Graph

Education

Income Longevity

1 2

Longevity Income

Education

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 ei ej (pairwise independence)

- no variance is zero

Structural Equation Model:

V = BV + E

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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

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Instantiated Models

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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

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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.

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Estimation

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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

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Hypothesis 2

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Updating

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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