**tl;dr**:These are the notes from a reading group I lead in 2016 on causal DAGs;
When I have time to expand these notes into complete sentences,
I will migrate the good bits to an expanded and improved
notebook on causal DAGS.

We will follow Pearl’s summary (J. Pearl (2009a)). (approx sections 1-3 of the Pearl paper.)

In particular, I want to get to the identification of causal effects given an existing causal DAG from observational data with unobserved covariates via criteria such as the back-door criterion We’ll see.

Approach: casual, motivate Pearl’s pronouncements, without deriving everything from axioms. Not statistical; will not answer the question of how we infer graph structure from data. Will skip many complexities by taking several slightly over-restrictive conditions, which we would relax if we were not doing this in 1 hour.

Not covered: UGs, PDAGs…

Assumptions: No-one here is an expert in this DAG graphical formalism for causal inference.

## Motivational examples

- Wet pavements
- Obesity contagion
- Nobel prizes and chocolate
- Simpson’s paradox
- etc

## Machinery

We are interested in representing influence between variables in a non-parametric fashion.

Our main tool to do this will be conditional independence DAGs, and causal use of these. Alternative name: “Bayesian Belief networks”. (Overloads “Bayesian”, so not used here)

### DAGs

DAG: Directed (probabilistic) graphical model. Graph defined, as usual, defined by a set of vertexes and edges.

\[ \mathcal{G}=(\mathbf{V},E) \]

We show the directs of edges by writing them as arrows.

For nodes \(X,Y\in V\) we write (XY) to mean there is a directed edged joining them.

Familiar from, e.g., Structural equation models, hierarchical models, expert systems. General graph theory…

A graph with *directed* edges, and no cycles.
(you cannot return to the same starting node traveling only *forward* along the arrows)

We need some terminology.

- Parents
- The parents of a node \(X\) in a graph are all nodes joined to it by an incoming arrow, \(\operatorname{parents}(X)=\{Y\in V:Y\rightarrow X\}.\)
- Children
- similarly, \(\operatorname{parents}(X)=\{Y\in V:X\rightarrow Y\}.\)
- Co-parent
- \(\operatorname{coparents}(X)=\{Y\in V:\exists Z\in V \text{ s.t. } X\rightarrow Z\text{ and }Y\rightarrow Z\}.\)

*Ancestors* and *descendants* should be clear as well.
For convenience, we define \(X\in\operatorname{parents}(X)\)

### Random variables

I will deal with finite collections of random variables \(\mathbf{V}\).

For simplicity of exposition, each of the RVs will be supported on \(\mathcal{X}_i\subset\mathbb{Z}\), so that we may work with pmfs, and write \(p(X_i|X_j)\) for the pmf. I may write \(p(x_i|x_j)\) to mean \(p(X_i=x_i|X_j=x_j)\).

Also we are working with *sets of random variables* rather than
*sets of events* and the discrete state space
reduces the need to discuss sets of events.

Extension to continuous RVs, or arbitrary RVs is trivial for everything I discuss here. (A challenge is if the probabilities are not all strictly positive.)

Motivation in terms of structural models.

\[ X_6 &= f_6(X_4, X_3, \varepsilon_6) \\ X_5 &= f_5(X_4, X_3, \varepsilon_5) \\ X_4 &= f_4(X_3, X_2, X_1, \varepsilon_4) \\ X_3 &= f_3(\varepsilon_3) \\ X_2 &= f_2(X_1, \varepsilon_2) \\ X_1 &= f_3(\varepsilon_1) \\ \]

Without further information about the forms of \(f_i\) or \(\varepsilon_i\), our assumptions have constrained our conditional independence relations to permit a particular factorization of the mass function:

\[ p(x_6, x_5, x_4, x_3, x_2, x_1) = p(x_1) p(x_2|x_1) p(x_3) p(x_4|x_1, x_2, x_3) p(x_5|x_3,x_4) p(x_6|x_3,x_4) \]

We are “nonparametric” in the sense that working with this conditional factorization does not require any further parametric assumptions on the model.

However, we would like to proceed from this factorization to conditional independence, which is non-trivial. Specifically, we would like to know which variables are conditionally independent of others, given such an (assumed) factorization.

More notation: We write

\[ X \perp Y|Z \]

for \(X\) independent of \(Y\) given \(Z\).

We also use this notation for sets of random variables, and will bold them when it is necessary to emphasis this.

\[ \mathbf{X} \perp \mathbf{Y}|\mathbf{Z} \]

Questions:

- \(X_2\perp X_3\)?
- \(X_2\perp X_3|X_1\)?
- \(X_2\perp X_3|X_4\)?

However, this product notation is not illuminating; we use a graph formalism. That’s where the DAGs come in.

This will proceed in 3 steps

The graphs will describe

*conditional factorization relations*.We will do some work to construct from these relations some

*conditional independence relations*, which may be read off the graph.From these relations plus a causal interpretation we will derive rules for

*identification of causal relations*If we get further than that, it will be all about coffee

Anyway, a joint distribution \(p(\mathbb{X})\) decomposes according to a directed graph \(G\) if we may factor it

\[ p(X_1,X_2,\dots,X_v)=\prod_{X=1}^v p(X_i|\operatorname{parents}(X_i)) \]

Uniqueness?

It would be tempting to suppose that a node is independent of its children given its parents or somesuch. But things are not quite so simple.

Questions:

- \(\text{Sprinkler}\perp \text{Rain}\)?
- \(\text{Sprinkler}\perp \text{Rain}|\text{Wet season}\)?
- \(\text{Sprinkler}\perp \text{Rain}|\text{Wet pavement}\)?
- \(\text{Sprinkler}\perp \text{Rain}|\text{Wet season}, \text{Wet pavement}\)?

To make precise statements about conditional independence relations we will do more work.

We need new graph vocabulary *and* conditional independence vocabulary.

Axiomatic characterisation of conditional independence. (Pearl 2008; Lauritzen 1996).

**Theorem**: ((Pearl 2008))
For disjoint subsets \(\mathbf{W},\mathbf{X},\mathbf{Y},\mathbf{Z}\subseteq\mathbf{V}.\)

Then the relation \(\cdot\perp\cdot|\cdot\) satisfies the following relations:

\[ \mathbf{X} \perp \mathbf{Z} |\mathbf{Y} & \Leftrightarrow & \mathbf{Z}\perp \mathbf{X} | \mathbf{Y} && \text{ Symmetry }&\\ \mathbf{X} \perp \mathbf{Y}\cup \mathbf{W} |\mathbf{Z} & \Rightarrow & \mathbf{X} \perp \mathbf{Y} \text{ and } \mathbf{X} \perp \mathbf{W} && \text{ Decomposition }&\\ \mathbf{X} \perp \mathbf{Y}\cup \mathbf{W} |\mathbf{Z} & \Rightarrow & \mathbf{X} \perp \mathbf{Y}|\mathbf{Z}\cup\mathbf{W} && \text{ Weak Union }&\\ \mathbf{X} \perp \mathbf{Y} |\mathbf{Z} \text{ and } \mathbf{X} \perp \mathbf{W}|\mathbf{Z}\cup \mathbf{Y} |\mathbf{Z} & \Rightarrow & \mathbf{X} \perp \mathbf{Y}\cup \mathbf{W}|\mathbf{Z} && \text{ Contraction }&\\ \mathbf{X} \perp \mathbf{Y} |\mathbf{Z}\cup \mathbf{W} \text{ and } \mathbf{X} \perp \mathbf{W} |\mathbf{Z}\cup \mathbf{Y} & \Rightarrow & \mathbf{X}\perp \mathbf{W}\cup\mathbf{Y} | \mathbf{Z} && \text{ Intersection } & (*)\\ \]

(*) The Intersection axiom only holds for strictly positive distributions.

(Pearl 2008):

How can we relate this to the topology of the graph?

The flow of conditional information does not correspond exactly to the marginal factorization, but it relates. (mention UG connections?)

**Definition**:
A set \(\mathbf{S}\) *blocks* a path \(\pi\) from X to Y in a DAG \(\mathcal{G}\) if either

There a node \(a\in\pi\) which

*is not*a collider on \(\pi\) such that \(a\in\mathbf{S}\)There a node \(b\in\pi\) which

*is*a collider on \(\pi\) and \(\operatorname{descendants}(b)\cap\mathbf{S}=\emptyset\)

If a path is not blocked, it is *active*.

**Definition**:
A set \(\mathbf{S}\) *d-separates* two subsets of nodes
\(\mathbf{X},\mathbf{X}\subseteq\mathcal{G}\)
if it blocks *every* path between any every pair of nodes \((A,B)\)
such that \(A\in\mathbf{X},\, B\in\mathbf{Y}.\)

This looks ghastly and unintuitive, but we have to live with it because it is the shortest path to making simple statements about conditional independence DAGs without horrible circumlocutions, or starting from undirected graphs, which is tedious.

**Theorem**: (Pearl 2008; Lauritzen 1996).
If the joint distribution of \(\mathbf{V}\) factorises according to the DAG
\(\mathbf{G}\) then for two subsets of variables
\(\mathbf{X}\perp\mathbf{Y}|\mathbf{S}\) iff \(\mathbf{S}\) *d*-separates \(\mathbf{X}\) and \(\mathbf{Y}\).

This puts us in a position to make non-awful, more intuitive statements about the conditional independence relationships that we may read off the DAG.

**Corollary**:
The DAG Markov property.

\[ X \perp \operatorname{descendants}(X)^C|\operatorname{parents}(X) \]

**Corollary**:
The DAG Markov blanket.

Define

\[ \operatorname{blanket}(X):= \operatorname{parents}(X)\cup \operatorname{children}(X)\cup \operatorname{coparents}(X) \]

Then

\[ X\perp \operatorname{blanket}(X)^C|\operatorname{blanket}(X) \]

## Causal interpretation

Finally!

We have a DAG \(\mathcal{G}\) and a set of variables \(\mathbf{V}\) to which we wish to give a causal interpretation.

Assume

- The \(\mathbf{V}\) factors according to \(\mathcal{G}\)
- \(X\rightarrow Y\) means “causes” (The Causal Markov property)
- We additionally assume
*faithfulness*, that is, that \(X\leftrightsquigarrow Y\) iff there is a path connecting them.

So, are we done? Only if correlation equals causation.

(Messerli 2012):

We add the additional condition that

- all the relevant variables are included in the graph. (We coyly avoid making this precise)

The BBC raised on possible confounding variable:

[…] Eric Cornell, who won the Nobel Prize in Physics in 2001, told Reuters “I attribute essentially all my success to the very large amount of chocolate that I consume. Personally I feel that milk chocolate makes you stupid… dark chocolate is the way to go. It’s one thing if you want a medicine or chemistry Nobel Prize but if you want a physics Nobel Prize it pretty much has got to be dark chocolate.”

Finally, we need to discuss the relationship between conditional dependence and causal effect. This is the difference between, say,

\[ P(\text{Wet pavement}|\text{Sprinkler}=on) \]

and

\[ P(\text{Wet pavement}|\operatorname{do}(\text{Sprinkler}=on)) \]

Called “truncated factorization” in the paper. Do-calculus and graph surgery.

If we know \(P\), this is relatively easy. Marginalize out all influences to the causal variable of interest, which we show graphically as wiping out a link.

Now suppose we are not given complete knowledge of \(P\), but only *some* of the conditional distributions. (there are *unobservable variables*).
This is the setup of observational studies and epidemiology and so on.

What variables *must* we know the conditional distributions of in order to know the conditional effect? That is, we call a set of covariates \(\mathbf{S}\) an *admissible set* (or *sufficient set*)
with respect to identifying the effect of \(X\) on \(Y\) iff

\[ p(Y=y|do(X=x))=\sum_{\mathbf{s}} P(Y=y|X=x,\mathbf{S}=\mathbf{s}) P(\mathbf{S}=\mathbf{s}) \]

**Criterion 1**:
The parents of a cause are an admissible set (J. Pearl 2009a)

**Criterion 2**:
The back door criterion.

A set \(\mathbf{S}\) such that

\(\mathbf{S}\cap\operatorname{descendants}(X)=\emptyset\)

\(\mathbf{S}\) blocks all paths which start with an arrow

*into*\(\mathbf{X}\)

This is a sufficient condition.

Causal properties of sufficient sets:

\[ P(Y=y|\operatorname{do}(X=x),S=s)=P(Y=y|X=x,S=s) \]

Hence

\[ P(Y=y|\operatorname{do}(X=x),S=s)=\sum_sP(Y=y|X=x,S=s)P(S=s) \]

## Examples

Social influence model in Shalizi and Thomas (2011):

- \(i,j\) are individuals,
- \(Z\) denote observed traits,
- \(X\) denote latent traits
- \(Y\) denote observed outcomes
- \(A\) is a network tie

\(X_i\) d-separates \(Y_i(t)\) from \(A_{ij}\). Since \(X_i\) is latent and unobserved, \(Y_i(t) \leftarrow X_i \rightarrow A_{ij}\) is a confounding path from \(Y_i(t)\) to \(A_{ij}\). Likewise \(Y_j(t-1)\leftarrow X_j \rightarrow A_{ij}\) is a confounding path from \(Yi(t-1)\) to \(A_{ij}\). Thus, \(Y_i(t)) and \(Y_i(t-1)\) are

d-connected when conditioning on all the observed (boxed) variables […] . Hence the direct effect of \(Y_i(t)\) on \(Y_i(t-1)\) is not identifiable

## Handy links

## Bonus bits

Equivalently we could do this:

## Recommended reading

People recommend me Koller and Friedman, which includes many different flavours of DAG model and many different methods, (Koller and Friedman 2009) but it didn’t suit me, being somehow too detailed and too non-specific at the same time.

Spirtes et al (Spirtes, Glymour, and Scheines 2001) and Pearl (J. Pearl 2009a) are readable. see also Pearl’s edited highlights (J. Pearl 2009b). Lauritzen ((Lauritzen 1996)) is clear but the details of the constructions are long and detailed and more general than here. (partially directed graphs.)

Lauritzen’s shorter introduction (Lauritzen 2000) is nice if you can get it; Not overwhelming, starts with a slightly more general formalism (DAGs as a special case of PDAGs, moral graphs everywhere). Murphy’s textbook (Murphy 2012) has a minimal introduction intermingled with some related models, with a more ML, “expert systems”-flavoured and more Bayesian formalism.

Aral, Sinan, Lev Muchnik, and Arun Sundararajan. 2009. “Distinguishing Influence-Based Contagion from Homophily-Driven Diffusion in Dynamic Networks.” *Proceedings of the National Academy of Sciences* 106 (51): 21544–9. https://doi.org/10.1073/pnas.0908800106.

Arnold, Barry C., Enrique Castillo, and Jose M. Sarabia. 1999. *Conditional Specification of Statistical Models*. Springer Science & Business Media. https://books.google.com.au/books?hl=en&lr=&id=lKeKu_HtMdQC&oi=fnd&pg=PA1&dq=arnold+castillo+sarabia+conditional+specification+of+statistical+models&ots=gxWoVEdsde&sig=p0BJlEeB5yQ052m5YhfQ_A6Kmoo.

Bareinboim, Elias, and Judea Pearl. 2016. “Causal Inference and the Data-Fusion Problem.” *Proceedings of the National Academy of Sciences* 113 (27): 7345–52. https://doi.org/10.1073/pnas.1510507113.

Bareinboim, Elias, Jin Tian, and Judea Pearl. 2014. “Recovering from Selection Bias in Causal and Statistical Inference.” In *AAAI*, 2410–6. http://ftp.cs.ucla.edu/pub/stat_ser/r425.pdf.

Bloniarz, Adam, Hanzhong Liu, Cun-Hui Zhang, Jasjeet Sekhon, and Bin Yu. 2015. “Lasso Adjustments of Treatment Effect Estimates in Randomized Experiments,” July. http://arxiv.org/abs/1507.03652.

Brodersen, Kay H., Fabian Gallusser, Jim Koehler, Nicolas Remy, and Steven L. Scott. 2015. “Inferring Causal Impact Using Bayesian Structural Time-Series Models.” *The Annals of Applied Statistics* 9 (1): 247–74. https://doi.org/10.1214/14-AOAS788.

Bühlmann, Peter. 2013. “Causal Statistical Inference in High Dimensions.” *Mathematical Methods of Operations Research* 77 (3): 357–70. https://doi.org/10.1007/s00186-012-0404-7.

Bühlmann, Peter, Markus Kalisch, and Lukas Meier. 2014. “High-Dimensional Statistics with a View Toward Applications in Biology.” *Annual Review of Statistics and Its Application* 1 (1): 255–78. https://doi.org/10.1146/annurev-statistics-022513-115545.

Bühlmann, Peter, Jonas Peters, Jan Ernest, and Marloes Maathuis. 2014. “Predicting Causal Effects in High-Dimensional Settings.” http://springmeeting2014.sfds.asso.fr/wp-content/uploads/2014/04/buhlmann.pdf.

Bühlmann, Peter, Philipp Rütimann, and Markus Kalisch. 2013. “Controlling False Positive Selections in High-Dimensional Regression and Causal Inference.” *Statistical Methods in Medical Research* 22 (5): 466–92. http://smm.sagepub.com/content/22/5/466.short.

Chen, B, and J Pearl. 2012. “Regression and Causation: A Critical Examination of Econometric Textbooks.”

Claassen, Tom, Joris M. Mooij, and Tom Heskes. 2014. “Proof Supplement - Learning Sparse Causal Models Is Not NP-Hard (UAI2013),” November. http://arxiv.org/abs/1411.1557.

Colombo, Diego, Marloes H. Maathuis, Markus Kalisch, and Thomas S. Richardson. 2012. “Learning High-Dimensional Directed Acyclic Graphs with Latent and Selection Variables.” *The Annals of Statistics* 40 (1): 294–321. http://projecteuclid.org/euclid.aos/1333567191.

De Luna, Xavier, Ingeborg Waernbaum, and Thomas S. Richardson. 2011. “Covariate Selection for the Nonparametric Estimation of an Average Treatment Effect.” *Biometrika*, October, asr041. https://doi.org/10.1093/biomet/asr041.

Elwert, Felix. 2013. “Graphical Causal Models.” In *Handbook of Causal Analysis for Social Research*, 245–73. Springer. http://publicifsv.sund.ku.dk/~sr/MPH/dag3/DAGs/Elwert_Chapter_2013_DAGs.pdf.

Ernest, Jan, and Peter Bühlmann. 2014. “Marginal Integration for Fully Robust Causal Inference,” May. http://arxiv.org/abs/1405.1868.

Gelman, Andrew. 2010. “Causality and Statistical Learning.” *American Journal of Sociology* 117 (3): 955–66. https://doi.org/10.1086/662659.

Hinton, Geoffrey E., Simon Osindero, and Kejie Bao. 2005. “Learning Causally Linked Markov Random Fields.” In *Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics*, 128–35. Citeseer. http://www.cs.toronto.edu/~osindero/PUBLICATIONS/HintonOsinderoBao05_CLMRF.pdf.

Jordan, Michael Irwin. 1999. *Learning in Graphical Models*. Cambridge, Mass.: MIT Press.

Jordan, Michael I., and Yair Weiss. 2002a. “Graphical Models: Probabilistic Inference.” *The Handbook of Brain Theory and Neural Networks*, 490–96. http://www.cs.iastate.edu/~honavar/jordan2.pdf.

———. 2002b. “Probabilistic Inference in Graphical Models.” *Handbook of Neural Networks and Brain Theory*. http://mlg.eng.cam.ac.uk/zoubin/course03/hbtnn2e-I.pdf.

Kalisch, Markus, and Peter Bühlmann. 2007. “Estimating High-Dimensional Directed Acyclic Graphs with the PC-Algorithm.” *Journal of Machine Learning Research* 8 (May): 613–36. http://jmlr.org/papers/v8/kalisch07a.html.

Kennedy, Edward H. 2015. “Semiparametric Theory and Empirical Processes in Causal Inference.” *arXiv Preprint arXiv:1510.04740*. http://arxiv.org/abs/1510.04740.

Kim, Jin H., and Judea Pearl. 1983. “A Computational Model for Causal and Diagnostic Reasoning in Inference Systems.” In *IJCAI*, 83:190–93. Citeseer. http://ijcai.org/Past%20Proceedings/IJCAI-83-VOL-1/PDF/041.pdf.

Koller, Daphne, and Nir Friedman. 2009. *Probabilistic Graphical Models : Principles and Techniques*. Cambridge, MA: MIT Press.

Lauritzen, S. L., and D. J. Spiegelhalter. 1988. “Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems.” *Journal of the Royal Statistical Society. Series B (Methodological)* 50 (2): 157–224. http://intersci.ss.uci.edu/wiki/pdf/Lauritzen1988.pdf.

Lauritzen, Steffen L. 2000. “Causal Inference from Graphical Models.” In *Complex Stochastic Systems*, 63–107. CRC Press. https://books.google.ch/books?hl=en&lr=&id=gCENL6qflA8C&oi=fnd&pg=PA63&ots=vgUI_QIs0y&sig=4WEKa7ToKKqHC1fsSt5prFZSL4Q.

———. 1996. *Graphical Models*. Clarendon Press.

Maathuis, Marloes H., and Diego Colombo. 2013. “A Generalized Backdoor Criterion.” *arXiv Preprint arXiv:1307.5636*. http://arxiv.org/abs/1307.5636.

Maathuis, Marloes H., Diego Colombo, Markus Kalisch, and Peter Bühlmann. 2010. “Predicting Causal Effects in Large-Scale Systems from Observational Data.” *Nature Methods* 7 (4): 247–48. https://doi.org/10.1038/nmeth0410-247.

Maathuis, Marloes H., Markus Kalisch, and Peter Bühlmann. 2009. “Estimating High-Dimensional Intervention Effects from Observational Data.” *The Annals of Statistics* 37 (6A): 3133–64. https://doi.org/10.1214/09-AOS685.

Marbach, Daniel, Robert J. Prill, Thomas Schaffter, Claudio Mattiussi, Dario Floreano, and Gustavo Stolovitzky. 2010. “Revealing Strengths and Weaknesses of Methods for Gene Network Inference.” *Proceedings of the National Academy of Sciences* 107 (14): 6286–91. https://doi.org/10.1073/pnas.0913357107.

Messerli, Franz H. 2012. “Chocolate Consumption, Cognitive Function, and Nobel Laureates.” *New England Journal of Medicine* 367 (16): 1562–4. https://doi.org/10.1056/NEJMon1211064.

Mihalkova, Lilyana, and Raymond J. Mooney. 2007. “Bottom-up Learning of Markov Logic Network Structure.” In *Proceedings of the 24th International Conference on Machine Learning*, 625–32. ACM. http://dl.acm.org/citation.cfm?id=1273575.

Montanari, Andrea. 2011. “Lecture Notes for Stat 375 Inference in Graphical Models.” http://www.stanford.edu/~montanar/TEACHING/Stat375/handouts/notes_stat375_1.pdf.

Murphy, Kevin P. 2012. *Machine Learning: A Probabilistic Perspective*. 1 edition. Adaptive Computation and Machine Learning Series. Cambridge, MA: MIT Press.

Neapolitan, Richard E., and others. 2004. *Learning Bayesian Networks*. Vol. 38. Prentice Hall Upper Saddle River. https://books.secure-services.me/Gentoomen%20Library/Artificial%20Intelligence/Bayesian%20networks/Learning%20Bayesian%20Networks%20-%20Neapolitan%20R.%20E..pdf.

Noel, Hans, and Brendan Nyhan. 2011. “The ‘Unfriending’ Problem: The Consequences of Homophily in Friendship Retention for Causal Estimates of Social Influence.” *Social Networks* 33 (3): 211–18. https://doi.org/10.1016/j.socnet.2011.05.003.

Pearl, Judea. 1982. “Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach.” In *In Proceedings of the National Conference on Artificial Intelligence*, 133–36. http://www.aaai.org/Papers/AAAI/1982/AAAI82-032.pdf.

———. 2008. *Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference*. Rev. 2. print., 12. [Dr.]. The Morgan Kaufmann Series in Representation and Reasoning. San Francisco, Calif: Kaufmann.

———. 2009a. “Causal Inference in Statistics: An Overview.” *Statistics Surveys* 3: 96–146. https://doi.org/10.1214/09-SS057.

———. 2009b. *Causality: Models, Reasoning and Inference*. Cambridge University Press.

———. 1986. “Fusion, Propagation, and Structuring in Belief Networks.” *Artificial Intelligence* 29 (3): 241–88. https://doi.org/10.1016/0004-3702(86)90072-X.

Peters, Jonas, Peter Bühlmann, and Nicolai Meinshausen. 2015. “Causal Inference Using Invariant Prediction: Identification and Confidence Intervals,” January. http://arxiv.org/abs/1501.01332.

Raginsky, M. 2011. “Directed Information and Pearl’s Causal Calculus.” In *2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)*, 958–65. https://doi.org/10.1109/Allerton.2011.6120270.

Rubin, Donald B, and Richard P Waterman. 2006. “Estimating the Causal Effects of Marketing Interventions Using Propensity Score Methodology.” *Statistical Science* 21 (2): 206–22. https://doi.org/10.1214/088342306000000259.

Sauer, Brian, and Tyler J. VanderWeele. 2013. *Use of Directed Acyclic Graphs*. Agency for Healthcare Research and Quality (US). https://www.ncbi.nlm.nih.gov/books/NBK126189/.

Shalizi, Cosma Rohilla, and Edward McFowland III. 2016. “Controlling for Latent Homophily in Social Networks Through Inferring Latent Locations,” July. http://arxiv.org/abs/1607.06565.

Shalizi, Cosma Rohilla, and Andrew C. Thomas. 2011. “Homophily and Contagion Are Generically Confounded in Observational Social Network Studies.” *Sociological Methods & Research* 40 (2): 211–39. https://doi.org/10.1177/0049124111404820.

Shpitser, Ilya, and Judea Pearl. 2008. “Complete Identification Methods for the Causal Hierarchy.” *The Journal of Machine Learning Research* 9: 1941–79.

Shpitser, Ilya, and Eric Tchetgen Tchetgen. 2014. “Causal Inference with a Graphical Hierarchy of Interventions,” November. http://arxiv.org/abs/1411.2127.

Smith, David A., and Jason Eisner. 2008. “Dependency Parsing by Belief Propagation.” In *Proceedings of the Conference on Empirical Methods in Natural Language Processing*, 145–56. Association for Computational Linguistics. http://dl.acm.org/citation.cfm?id=1613737.

Spirtes, Peter, Clark Glymour, and Richard Scheines. 2001. *Causation, Prediction, and Search*. Second Edition. Adaptive Computation and Machine Learning. The MIT Press. https://www.cs.cmu.edu/afs/cs.cmu.edu/project/learn-43/lib/photoz/.g/scottd/fullbook.pdf.

Vansteelandt, Stijn, Maarten Bekaert, and Gerda Claeskens. 2012. “On Model Selection and Model Misspecification in Causal Inference.” *Statistical Methods in Medical Research* 21 (1): 7–30. https://doi.org/10.1177/0962280210387717.

Wright, Sewall. 1934. “The Method of Path Coefficients.” *The Annals of Mathematical Statistics* 5 (3): 161–215. https://doi.org/10.1214/aoms/1177732676.

Yedidia, J. S., W. T. Freeman, and Y. Weiss. 2003. “Understanding Belief Propagation and Its Generalizations.” In *Exploring Artificial Intelligence in the New Millennium*, edited by G. Lakemeyer and B. Nebel, 239–36. Morgan Kaufmann Publishers. http://www.merl.com/publications/TR2001-22.

Zhang, Kun, Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. 2012. “Kernel-Based Conditional Independence Test and Application in Causal Discovery,” February. http://arxiv.org/abs/1202.3775.