Inferring cause and effect from nature. Graphical models and related techniques for doing it. Avoiding the danger of folk statistics. Observational studies, confounding, adjustment criteria, d-separation, identifiability, interventions, moral equivalence…
The most well-trodden path here is using directed graphical models with the additional assumption that \(A\rightarrow B\) may be read as “A causes a change in B”. C&C instrumental variables and propensity score matching. When you are talking Structural Equation models, this boils down to more or less some extra interpretation imposed on hierarchical models. Avoidance of Ecological fallacy/ Simpson’s paradox.
When can I use my crappy observational data, collected without a good experimental design for whatever reason, to do interventional inference? There is a lot of research in this. I should summarise the salient bits for myself. In fact I did; I led a reading group on this.
I speculate that in realistic causal networks or DAGs, the number of possible correlations grows faster than the number of possible causal relationships. So confounds really are that common, and since people do not think in DAGs, the imbalance also explains overconfidence.
Samantha Klinberg has a book notable for its handling for time-dependent causality.
Tutorial: David Sontag and Uri Shalit, Causal inference from observational studies.
Chapter 3 of (some edition of) Pearl’s book is available as an author’s preprint:
Rubin and Waterman (2006) comes recommended by Shalizi as:
A good description of Rubin et al.’s methods for causal inference, adapted to the meanest understanding. […] Rubin and Waterman do a very good job of explaining, in a clear and concrete problem, just how and why the newer techniques of causal inference are valuable, with just enough technical detail that it doesn’t seem like magic.
Causal Graph inference from data
Uh oh. You don’t know what causes what? Or specifically, you can’t eliminate a whole bunch of potential causal arrows a priori? Much more work.
Here is a seminar I noticed on this theme, which is also a lightspeed introduction to some difficulties.
Guido Consonni, Objective Bayes Model Selection of Gaussian Essential Graphs with Observational and Interventional Data.
Graphical models based on Directed Acyclic Graphs (DAGs) represent a powerful tool for investigating dependencies among variables. It is well known that one cannot distinguish between DAGs encoding the same set of conditional independencies (Markov equivalent DAGs) using only observational data. However, the space of all DAGs can be partitioned into Markov equivalence classes, each being represented by a unique Essential Graph (EG), also called Completed Partially Directed Graph (CPDAG). In some fields, in particular genomics, one can have both observational and interventional data, the latter being produced after an exogenous perturbation of some variables in the system, or from randomized intervention experiments. Interventions destroy the original causal structure, and modify the Markov property of the underlying DAG, leading to a finer partition of DAGs into equivalence classes, each one being represented by an Interventional Essential Graph (I-EG) (Hauser and Buehlmann). In this talk we consider Bayesian model selection of EGs under the assumption that the variables are jointly Gaussian. In particular, we adopt an objective Bayes approach, based on the notion of fractional Bayes factor, and obtain a closed form expression for the marginal likelihood of an EG. Next we construct a Markov chain to explore the EG space under a sparsity constraint, and propose an MCMC algorithm to approximate the posterior distribution over the space of EGs. Our methodology, which we name Objective Bayes Essential graph Search (OBES), allows to evaluate the inferential uncertainty associated to any features of interest, for instance the posterior probability of edge inclusion. An extension of OBES to deal simultaneously with observational and interventional data is also presented: this involves suitable modifications of the likelihood and prior, as well as of the MCMC algorithm. We conclude by presenting results for simulated and real experiments (protein-signaling data).
This is joint work with Federico Castelletti, Stefano Peluso and Marco Della Vedova (Universita' Cattolica del Sacro Cuore).
Causal time series DAGS
As with other time series methods, has its own issues.
The CausalImpact R package implements an approach to estimating the causal effect of a designed intervention on a time series. For example, how many additional daily clicks were generated by an advertising campaign? Answering a question like this can be difficult when a randomized experiment is not available. The package aims to address this difficulty using a structural Bayesian time-series model to estimate how the response metric might have evolved after the intervention if the intervention had not occurred.
Achab, Massil, Emmanuel Bacry, Stéphane Gaïffas, Iacopo Mastromatteo, and Jean-Francois Muzy. 2017. “Uncovering Causality from Multivariate Hawkes Integrated Cumulants.” In PMLR. http://arxiv.org/abs/1607.06333.
Allen, John-Mark A., Jonathan Barrett, Dominic C. Horsman, Ciarán M. Lee, and Robert W. Spekkens. 2017. “Quantum Common Causes and Quantum Causal Models.” Physical Review X 7 (3): 031021. https://doi.org/10.1103/PhysRevX.7.031021.
Aragam, Bryon, Jiaying Gu, and Qing Zhou. 2017. “Learning Large-Scale Bayesian Networks with the Sparsebn Package,” March. http://arxiv.org/abs/1703.04025.
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.
Bahadori, Mohammad Taha, Krzysztof Chalupka, Edward Choi, Robert Chen, Walter F. Stewart, and Jimeng Sun. 2017. “Neural Causal Regularization Under the Independence of Mechanisms Assumption,” February. http://arxiv.org/abs/1702.02604.
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.
Bonchi, Francesco, Francesco Gullo, Bud Mishra, and Daniele Ramazzotti. 2018. “Probabilistic Causal Analysis of Social Influence.” In Proceedings of the 27th ACM International Conference on Information and Knowledge Management, 1003–12. CIKM ’18. New York, NY, USA: ACM. https://doi.org/10.1145/3269206.3271756.
Bongers, Stephan, Jonas Peters, Bernhard Schölkopf, and Joris M. Mooij. 2016. “Structural Causal Models: Cycles, Marginalizations, Exogenous Reparametrizations and Reductions,” November. http://arxiv.org/abs/1611.06221.
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.
Chalak, Karim, and Halbert White. 2012. “Causality, Conditional Independence, and Graphical Separation in Settable Systems.” Neural Computation 24 (7): 1611–68. http://ieeexplore.ieee.org/abstract/document/6797310/.
Chaves, Rafael, Gabriela Barreto Lemos, and Jacques Pienaar. 2018. “Causal Modeling the Delayed-Choice Experiment.” Physical Review Letters 120 (19): 190401. https://doi.org/10.1103/PhysRevLett.120.190401.
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.
Didelez, Vanessa. n.d. “Causal Reasoning for Events in Continuous Time: A Decision–Theoretic Approach.” Accessed July 18, 2015. http://www.homepages.ucl.ac.uk/~ucgtrbd/uai2015_causal/papers/didelez.pdf.
Duvenaud, David K., Daniel Eaton, Kevin P. Murphy, and Mark W. Schmidt. 2010. “Causal Learning Without DAGs.” In NIPS Causality: Objectives and Assessment, 177–90. http://jmlr.org/proceedings/papers/v6/duvenaud10a/duvenaud10a.pdf.
Eichler, Michael. 2001. “Granger-Causality Graphs for Multivariate Time Series.” Granger-Causality Graphs for Multivariate Time Series. http://archiv.ub.uni-heidelberg.de/volltextserver/20749/1/beitrag.64.pdf.
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.
Entner, Doris, Patrik Hoyer, and Peter Spirtes. 2013. “Data-Driven Covariate Selection for Nonparametric Estimation of Causal Effects.” In Proceedings of the Sixteenth International Conference on Artificial Intelligence and Statistics, 256–64. http://jmlr.org/proceedings/papers/v31/entner13a.html.
Ernest, Jan, and Peter Bühlmann. 2014. “Marginal Integration for Fully Robust Causal Inference,” May. http://arxiv.org/abs/1405.1868.
Fixx, James F. 1977. Games for the Superintelligent. London: Muller.
Fu, Fei, and Qing Zhou. 2013. “Learning Sparse Causal Gaussian Networks with Experimental Intervention: Regularization and Coordinate Descent.” Journal of the American Statistical Association 108 (501): 288–300. https://doi.org/10.1080/01621459.2012.754359.
Gebharter, Alexander, and Nina Retzlaff. 2020. “A New Proposal How to Handle Counterexamples to Markov Causation à La Cartwright, or: Fixing the Chemical Factory.” Synthese 197 (4): 1467–86. https://doi.org/10.1007/s11229-018-02014-7.
Gelman, Andrew. 2010. “Causality and Statistical Learning.” American Journal of Sociology 117 (3): 955–66. https://doi.org/10.1086/662659.
Geng, Zhi, Yue Liu, Chunchen Liu, and Wang Miao. 2019. “Evaluation of Causal Effects and Local Structure Learning of Causal Networks.” Annual Review of Statistics and Its Application 6 (1): 103–24. https://doi.org/10.1146/annurev-statistics-030718-105312.
Gu, Jiaying, Fei Fu, and Qing Zhou. 2014. “Adaptive Penalized Estimation of Directed Acyclic Graphs from Categorical Data,” March. http://arxiv.org/abs/1403.2310.
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.
Huang, Yuxiao, and Samantha Kleinberg. n.d. “Fast and Accurate Causal Inference from Time Series Data,” 6. http://www.skleinberg.org/papers/huang_flairs15.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.
Kilbertus, Niki, Mateo Rojas-Carulla, Giambattista Parascandolo, Moritz Hardt, Dominik Janzing, and Bernhard Schölkopf. 2017. “Avoiding Discrimination Through Causal Reasoning,” June. http://arxiv.org/abs/1706.02744.
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.
Kleinberg, Samantha. 2012. Causality, Probability, and Time. 1 edition. Cambridge: Cambridge University Press.
Kohler, Ulrich, Frauke Kreuter, and Elizabeth A. Stuart. 2019. “Nonprobability Sampling and Causal Analysis.” Annual Review of Statistics and Its Application 6 (1): 149–72. https://doi.org/10.1146/annurev-statistics-030718-104951.
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.
Lopez-Paz, David, Robert Nishihara, Soumith Chintala, Bernhard Schölkopf, and Léon Bottou. 2016. “Discovering Causal Signals in Images,” May. http://arxiv.org/abs/1605.08179.
Louizos, Christos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling. 2017. “Causal Effect Inference with Deep Latent-Variable Models.” In Advances in Neural Information Processing Systems 30, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 6446–56. Curran Associates, Inc. http://papers.nips.cc/paper/7223-causal-effect-inference-with-deep-latent-variable-models.pdf.
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.
Malinsky, Daniel, Ilya Shpitser, and Thomas Richardson. 2019. “A Potential Outcomes Calculus for Identifying Conditional Path-Specific Effects,” March. http://arxiv.org/abs/1903.03662.
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.
Pearl, Judea, and Elias Bareinboim. 2014. “External Validity: From Do-Calculus to Transportability Across Populations.” Statistical Science 29 (4): 579–95. https://doi.org/10.1214/14-STS486.
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/.
Schölkopf, Bernhard, Krikamol Muandet, Kenji Fukumizu, and Jonas Peters. 2015. “Computing Functions of Random Variables via Reproducing Kernel Hilbert Space Representations,” January. http://arxiv.org/abs/1501.06794.
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.
Textor, Johannes, Alexander Idelberger, and Maciej Liśkiewicz. 2015. “Learning from Pairwise Marginal Independencies,” August. http://arxiv.org/abs/1508.00280.
Tu, Ruibo, Cheng Zhang, Paul Ackermann, Hedvig Kjellström, and Kun Zhang. 2018. “Causal Discovery in the Presence of Missing Data,” July. http://arxiv.org/abs/1807.04010.
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.
Visweswaran, Shyam, and Gregory F. Cooper. 2014. “Counting Markov Blanket Structures,” July. http://arxiv.org/abs/1407.2483.
Walker, Jeffrey A. 2014. “The Effect of Unmeasured Confounders on the Ability to Estimate a True Performance or Selection Gradient (and Other Partial Regression Coefficients).” Evolution 68 (7): 2128–36. https://doi.org/10.1111/evo.12406.
Wright, Sewall. 1934. “The Method of Path Coefficients.” The Annals of Mathematical Statistics 5 (3): 161–215. https://doi.org/10.1214/aoms/1177732676.
Yadav, Pranjul, Lisiane Prunelli, Alexander Hoff, Michael Steinbach, Bonnie Westra, Vipin Kumar, and Gyorgy Simon. 2016. “Causal Inference in Observational Data,” November. http://arxiv.org/abs/1611.04660.
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.