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Graphical Models for Categorical Data
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Graphical Models for Categorical Data

For advanced students of network data science, this compact account covers both well-established methodology and the theory of models recently introduced in the graphical model literature. It focuses on the discrete case where all variables involved are categorical and, in this context, it achieves a unified presentation of classical and recent results.

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  • COPYRIGHT: © Alberto Roverato 2017
Agresti, A. (2013). Categorical Data Analysis, 3rd edn, New York: John Wiley and Sons.
Ali, R. A., Richardson, T. S. & Spirtes, P. (2009). Markov equivalence for ancestral graphs. The Annals of Statistics, 37(5B), 2808–37.
Anderson, T. W. (1969). Statistical inference for covariance matrices with linear structure. In Multivariate Analysis, II: Proc. 2nd Int. Symp., Dayton, Ohio, 1968. New York: Academic Press, pp. 5566.
Anderson, T. W. (1973). Asymptotically efficient estimation of covariance matrices with linear structure. The Annals of Statistics, 1(1), 135–41.
Andersson, S. A., Madigan, D., Perlman, M. D. (1997). A characterization of Markov equivalence classes for acyclic digraphs. The Annals of Statistics, 25(2), 505–41.
Andersson, S. A., Madigan, D. & Perlman, M. D. (2001). Alternative Markov properties for chain graphs. Scandinavian Journal of Statistics, 28(1), 3385.
Asmussen, S. & Edwards, D. (1983). Collapsibility and response variables in contingency tables. Biometrika, 70(3), 567–78.
Barber, D. (2012). Bayesian Reasoning and Machine Learning. Cambridge: Cambridge University Press.
Barndorff-Nielsen, O. (1978). Information in Exponential Families and Conditioning. New York: John Wiley and Sons.
Barndorff-Nielsen, O. (2014). Information and Exponential Families in Statistical Theory. Chichester: John Wiley and Sons.
Bartolucci, F., Colombi, R. & Forcina, A. (2007). An extended class of marginal link functions for modelling contingency tables by equality and inequality constraints. Statistica Sinica, 17(2), 691711.
Bergsma, W. P. & Rudas, T. (2002). Marginal models for categorical data. The Annals of Statistics, 30(1), 140–59.
Birch, M. W. (1963). Maximum likelihood in three-way contingency tables. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 25(1), 220233.
Bishop, Y. M., Fienberg, S. E. & Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. Cambridge, MA: MIT Press.
Bishop, Y. M., Fienberg, S. E. & Holland, P. W. (2007). Discrete Multivariate Analysis: Theory and Practice. New York: Springer-Verlag.
Boutilier, C., Friedman, N., Goldszmidt, M. & Koller, D. (1996). Context-specific independence in Bayesian networks.: Proceedings of the Twelfth Annual Conference on Uncertainty in Artificial Intelligence (UAI-96). San Francisco, CA: Morgan Kaufmann, pp. 115–23.
Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Lecture Notes-monograph series, vol. 9. Hayward, CA: Institute of Mathematical Statistics.
Chickering, D. M. (2002). Learning equivalence classes of Bayesian-network structures. Journal of Machine Learning Research, 2, 445–98.
Christensen, R. (1997). Log-linear Models and Logistic Regression, 2nd edn, New York: Springer-Verlag.
Consonni, G. & Leucari, V. (2006). Reference priors for discrete graphical models. Biometrika, 93(1), 2340.
Coppen, A. (1966). The Marke–Nyman temperament scale: an English translation. British Journal of Medical Psychology, 39(1), 55–9.
Corander, J. (2003). Labelled graphical models. Scandinavian Journal of Statistics, 30(3), 493508.
Cowell, R. G., Dawid, A. P., Lauritzen, S. L. & Spiegelhalter, D. J. (1999). Probabilistic Networks and Expert Systems. New York: Springer-Verlag.
Cox, D. R. & Wermuth, N. (1993). Linear dependencies represented by chain graphs. Statistical Science, 8(3), 204–18.
Cox, D. R. & Wermuth, N. (1996). Multivariate Dependencies: Models, Analysis, and Interpretation. Boca Raton, FL: Chapman & Hall.
Darroch, J. N. & Ratcliff, D. (1972). Generalized iterative scaling for log-linear models. The Annals of Mathematical Statistics, 43(5), 1470–80.
Darroch, J. N., Lauritzen, S. L. & Speed, T. P. (1980). Markov fields and log-linear interaction models for contingency tables. The Annals of Statistics, 8(3), 522–39.
Davison, A. C. (2003). Statistical Models. Vol. 11. Cambridge: Cambridge University Press.
Dawid, A. P. (1979). Conditional independence in statistical theory. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 41, 131.
Dawid, A. P. & Lauritzen, S. L. (1993). Hyper Markov laws in the statistical analysis of decomposable graphical models. The Annals of Statistics, 21(3), 1272–317.
Deming, W. E. & Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. The Annals of Mathematical Statistics, 11(4), 427–44.
Diestel, R. (1990). Graph Decompositions: A Study in Infinite Graph Theory. Oxford: Clarendon Press.
Drton, M. (2008). Iterative conditional fitting for discrete chain graph models. In Brito, P., ed., COMPSTAT 2008 – Proceedings in Computational Statistics. New York: Springer, pp. 93104.
Drton, M. (2009). Discrete chain graph models. Bernoulli, 15(3), 736–53.
Drton, M. & Maathuis, M. H. (2017). Structure learning in graphical modeling. Annual Review of Statistics and Its Application, 4(1).
Drton, M. & Richardson, T. S. (2008a). Binary models for marginal independence. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(2), 287309.
Drton, M. & Richardson, T. S. (2008b). Graphical methods for efficient likelihood inference in Gaussian covariance models. Journal of Machine Learning Research, 9, 893914.
Drton, M., Lauritzen, S. L., Maathuis, M. & Wainwright, M. (2017). Handbook of Graphical Models. Boca Raton, FL: Chapman and Hall/CRC.
Edwards, D. (2000). Introduction to Graphical Modelling, 2nd edn, New York: Springer-Verlag.
Edwards, D. & Kreiner, S. (1983). The analysis of contingency tables by graphical models. Biometrika, 70(3), 553–65.
Evans, R. J. (2016). Graphs for margins of Bayesian networks. Scandinavian Journal of Statistics, 43(3), 625–48.
Evans, R. J. & Forcina, A. (2013). Two algorithms for fitting constrained marginal models. Computational Statistics & Data analysis, 66, 17.
Evans, R. J. & Richardson, T. S. (2010). Maximum likelihood fitting of acyclic directed mixed graphs to binary data. In Grunwald, P. & Spirtes, P., eds, Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence (UAI 2010). Corvallis, OR: AUAI Press, pp. 177–84.
Evans, R. J. & Richardson, T. S. (2013). Marginal log-linear parameters for graphical Markov models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4), 743–68.
Evans, R. J. & Richardson, T. S. (2014). Markovian acyclic directed mixed graphs for discrete data. The Annals of Statistics, 42(4), 1452–82.
Frydenberg, M. (1990). The chain graph Markov property. Scandinavian Journal of Statistics, 17(4), 333353.
Frydenberg, M. & Lauritzen, S. L. (1989). Decomposition of maximum likelihood in mixed graphical interaction models. Biometrika, 76(3), 539–55.
Geiger, D. & Meek, C. (1998). Graphical models and exponential families. In Proceedings of the Fourteenth Annual Conference on Uncertainty in Artificial Intelligence (UAI-98). San Francisco, CA: Morgan Kaufmann, pp. 156–65.
Geiger, D. & Pearl, J. (1988). On the logic of causal models. In Uncertainty in Artificial Intelligence 4 Annual Conference on Uncertainty in Artificial Intelligence (UAI-88). Amsterdam: Elsevier Science, pp. 314.
Geiger, D. & Pearl, J. (1993). Logical and algorithmic properties of conditional independence and graphical models. The Annals of Statistics, 21(4), 2001–21.
Graybill, F. A. (1983). Matrices with Applications in Statistics. Belmont, CA: Wadsworth.
Gutiérrez-Peña, E. & Smith, A. F. M. (1997). Exponential and Bayesian conjugate families: review and extensions. Test, 6(1), 190.
Hall, P. (1934). A contribution to the theory of groups of prime-power order. Proceedings of the London Mathematical Society, 2(1), 2995.
Hammersley, J. M. & Clifford, P. (1971). Markov fields on finite graphs and lattices. Unpublished manuscript.
Højsgaard, S. (2004). Statistical inference in context specific interaction models for contingency tables. Scandinavian Journal of Statistics, 31(1), 143–58.
Højsgaard, S., Edwards, D. & Lauritzen, S. L. (2012). Graphical Models with R. New York: Springer Science+Business Media.
Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 186(1007), 453–61.
Jeffreys, H. (1961). Theory of Probability, 3rd edn. Oxford Classic Texts in the Physical Sciences. Oxford: Oxford University Press.
Jokinen, J. (2006). Fast estimation algorithm for likelihood-based analysis of repeated categorical responses. Computational Statistics & Data Analysis, 51(3), 1509–22.
Kass, R. E. & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–95.
Kauermann, G. (1996). On a dualization of graphical Gaussian models. Scandinavian Journal of Statistics, 23(1), 105–16.
Kauermann, G. (1997). A note on multivariate logistic models for contingency tables. Australian Journal of Statistics, 39(3), 261–76.
Koski, T. and Noble, J. M. (2009). Graphical models and exponential families. In Bayesian Networks: An Introduction. Chichester: John Wiley and Sons, Ltd, chapter 8.
La Rocca, L. & Roverato, A. (2017). Discrete graphical models. In Drton, M., Lauritzen, S. L., Maathuis, M. & Wainwright, M., eds, Handbook of Graphical Models. Handbooks of Modern Statistical Methods. Boca Raton, FL: Chapman and Hall/CRC.
Lang, J. B. (1996. Maximum likelihood methods for a generalized class of log-linear models. The Annals of Statistics, 24(2), 726–52.
Lauritzen, S. L. (1996). Graphical models. Oxford: Clarendon Press.
Lauritzen, S. L. (2001). Causal inference from graphical models. In Barndorff-Nielsen, O.E., Cox, D. R. & Klüppelberg, C., eds, Complex Stochastic Systems. London/Boca Raton: Chapman and Hall/CRC Press, pp. 63107.
Lauritzen, S. L. & Richardson, T. S. (2002). Chain graph models and their causal interpretations. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(3), 321–48.
Lauritzen, S. L. & Wermuth, N. (1989). Graphical models for associations between variables, some of which are qualitative and some quantitative. The Annals of Statistics, 17(1), 3157.
Lauritzen, S. L., Dawid, A. P., Larsen, B. N. & Leimer, H.-G. (1990). Independence properties of directed Markov fields. Networks, 20(5), 491505.
Lovász, L. (1993). Combinatorial Problems and Exercises, 2nd edn. Amsterdam: North-Holland.
Lupparelli, M. & Roverato, A. (2017). Log-mean linear regression models for binary responses with an application to multimorbidity. Journal of the Royal Statistical Society: Series C (Applied Statistics), 66(2), 227252.
Lupparelli, M., Marchetti, G. M. & Bergsma, W. P. (2009). Parameterizations and fitting of bi-directed graph models to categorical data. Scandinavian Journal of Statistics, 36(3), 559–76.
Lütkepol, H. (1996). Handbook of Matrices. Chichester: Wiley.
Madsen, M. (1976). Statistical analysis of multiple contingency tables. Two examples. Scandinavian Journal of Statistics, 3(3), 97106.
Marchetti, G. M. & Lupparelli, M. (2011). Chain graph models of multivariate regression type for categorical data. Bernoulli, 17(3), 827–44.
Massam, H., Liu, J. & Dobra, A. (2009). A conjugate prior for discrete hierarchical log-linear models. The Annals of Statistics, 37(6), 3431–67.
Meek, C. (1995). Causal inference and causal explanation with background knowledge. In Proceedings of the Eleventh Annual Conference on Uncertainty in Artificial Intelligence (UAI-95). San Francisco, CA: Morgan Kaufmann, pp. 403–10.
Morris, C. N. (1982). Natural exponential families with quadratic variance functions. The Annals of Statistics, 10(1), 6580.
Morris, C. N. (1983). Natural exponential families with quadratic variance functions: statistical theory. The Annals of Statistics, 11(2), 515–29.
Nyman, H., Pensar, J., Koski, T. & Corander, J. (2014). Stratified graphical models – context-specific independence in graphical models. Bayesian Analysis, 9(4), 883908.
Pearl, J. (1986). Fusion, propagation, and structuring in belief networks. Artificial Intelligence, 29(3), 241–88.
Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufmann.
Pearl, J. (2009). Causality, 2nd edn, Cambridge: Cambridge University Press.
Pearl, J. & Paz, A. (1987). Graphoids: a graph-based logic for reasoning about relevancy relations. In Boulary, B. D., Hogg, D. &Steel, L., eds, Advances in Artificial Intelligence – II. Amsterdam: North-Holland, pp. 357–63.
Pearl, J. & Verma, T. (1990). Equivalence and synthesis of causal models. In Uncertainty in Artificial Intelligence 6 Annual Conference on Uncertainty in Artificial Intelligence (UAI-90). Amsterdam: Elsevier Science, pp. 255–68.
Piccioni, M. (2000). Independence structure of natural conjugate densities to exponential families and the Gibbs’ sampler. Scandinavian Journal of Statistics, 27(1), 111–27.
R Core Team. (2016). R: A Language and Environment for Statistical Computing. Vienna: Foundation for Statistical Computing.
Richardson, T. & Spirtes, P. (2002). Ancestral graph Markov models. The Annals of Statistics, 30(4), 9621030.
Richardson, T. S. (2003). Markov properties for acyclic directed mixed graphs. Scandinavian Journal of Statistics, 30(1), 145–57.
Rota, G.-C. (1964). On the foundations of combinatorial theory I. Theory of Möbius functions. Probability Theory and Related Fields, 2(4), 340–68.
Roverato, A. (2005). A unified approach to the characterization of equivalence classes of DAGs, chain graphs with no flags and chain graphs. Scandinavian Journal of Statistics, 32(2), 295312.
Roverato, A. (2015). Log-mean linear parameterization for discrete graphical models of marginal independence and the analysis of dichotomizations. Scandinavian Journal of Statistics, 42(2), 627–48.
Roverato, A. & La Rocca, L. (2006). On block ordering of variables in graphical modelling. Scandinavian Journal of Statistics, 33(1), 6581.
Roverato, A. & Studenỳ, M. (2006). A graphical representation of equivalence classes of AMP chain graphs. Journal of Machine Learning Research, 7, 1045–78.
Roverato, A. & Whittaker, J. (1998). The Isserlis matrix and its application to non-decomposable graphical Gaussian models. Biometrika, 85(3), 711–25.
Roverato, A., Lupparelli, M. & La Rocca, L. (2013). Log-mean linear models for binary data. Biometrika, 100(2), 485–94.
Rudas, T., Bergsma, W. P. & Németh, R. (2010). Marginal log-linear parameterization of conditional independence models. Biometrika, 97(4), 1006–12.
Sadeghi, K. & Lauritzen, S. L. (2014). Markov properties for mixed graphs. Bernoulli, 20(2), 676–96.
Sadeghi, K. & Wermuth, N. (2016). Pairwise Markov properties for regression graphs. Stat, 5, 286–94.
Speed, T. P. (1983). Cumulants and partition lattices. Australian Journal of Statistics, 25(2), 378–88.
Spirtes, P., Glymour, C. & Scheines, R. (2000). Causation, Prediction, and Search, 2nd edn, Cambridge, MA: MIT Press.
Studenỳ, M. (2005). Probabilistic Conditional Independence Structures. London: Springer-Verlag.
Tarjan, R. E. & Yannakakis, M. (1984). Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13(3), 566–79.
Volf, M. & Studenỳ, M. (1999). A graphical characterization of the largest chain graphs. International Journal of Approximate Reasoning, 20(3), 209–36.
Weisner, L. (1935). Abstract theory of inversion of finite series. Transactions of the American Mathematical Society, 38(3), 474–84.
Wermuth, N. (1976). Model search among multiplicative models. Biometrics, 32(2), 253–63.
Wermuth, N. & Cox, D. R. (2015). Graphical Markov models: overview. In Wright, J. D., ed., International Encyclopedia of the Social and Behavioral Sciences, 2nd edn, vol. 10. Oxford: Elesevier, pp. 341–50.
Wermuth, N. & Lauritzen, S. L. (1990). On substantive research hypotheses, conditional independence graphs and graphical chain models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 52(1), 2150.
Wermuth, N. & Sadeghi, K. (2012). Sequences of regressions and their independences. TEST, 21(2), 215–52.
Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Chichester: Wiley.
Wright, S. (1921). Correlation and causation. Journal of Agricultural Research, 20(7), 557–85.

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