Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T06:29:35.183Z Has data issue: false hasContentIssue false
Series:   SemStat Elements

Graphical Models for Categorical Data

Published online by Cambridge University Press:  16 June 2017

Alberto Roverato
Alberto Roverato
Affiliation:
Università di Bologna

Summary

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.
Get access
Type
Element
Information
Series: SemStat Elements
Online ISBN: 9781108277495
Publisher: Cambridge University Press
Print publication: 24 August 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Bibliography

Agresti, A. (2013). Categorical Data Analysis, 3rd edn, New York: John Wiley and Sons.Google Scholar
Ali, R. A., Richardson, T. S. & Spirtes, P. (2009). Markov equivalence for ancestral graphs. The Annals of Statistics, 37(5B), 2808–37.CrossRefGoogle Scholar
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. 55–66.Google Scholar
Anderson, T. W. (1973). Asymptotically efficient estimation of covariance matrices with linear structure. The Annals of Statistics, 1(1), 135–41.CrossRefGoogle Scholar
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.Google Scholar
Andersson, S. A., Madigan, D. & Perlman, M. D. (2001). Alternative Markov properties for chain graphs. Scandinavian Journal of Statistics, 28(1), 33–85.Google Scholar
Asmussen, S. & Edwards, D. (1983). Collapsibility and response variables in contingency tables. Biometrika, 70(3), 567–78.CrossRefGoogle Scholar
Barber, D. (2012). Bayesian Reasoning and Machine Learning. Cambridge: Cambridge University Press.Google Scholar
Barndorff-Nielsen, O. (1978). Information in Exponential Families and Conditioning. New York: John Wiley and Sons.Google Scholar
Barndorff-Nielsen, O. (2014). Information and Exponential Families in Statistical Theory. Chichester: John Wiley and Sons.CrossRefGoogle Scholar
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), 691–711.Google Scholar
Bergsma, W. P. & Rudas, T. (2002). Marginal models for categorical data. The Annals of Statistics, 30(1), 140–59.CrossRefGoogle Scholar
Birch, M. W. (1963). Maximum likelihood in three-way contingency tables. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 25(1), 220–233.Google Scholar
Bishop, Y. M., Fienberg, S. E. & Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. Cambridge, MA: MIT Press.Google Scholar
Bishop, Y. M., Fienberg, S. E. & Holland, P. W. (2007). Discrete Multivariate Analysis: Theory and Practice. New York: Springer-Verlag.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Chickering, D. M. (2002). Learning equivalence classes of Bayesian-network structures. Journal of Machine Learning Research, 2, 445–98.Google Scholar
Christensen, R. (1997). Log-linear Models and Logistic Regression, 2nd edn, New York: Springer-Verlag.Google Scholar
Consonni, G. & Leucari, V. (2006). Reference priors for discrete graphical models. Biometrika, 93(1), 23–40.Google Scholar
Coppen, A. (1966). The Marke–Nyman temperament scale: an English translation. British Journal of Medical Psychology, 39(1), 55–9.Google Scholar
Corander, J. (2003). Labelled graphical models. Scandinavian Journal of Statistics, 30(3), 493–508.CrossRefGoogle Scholar
Cowell, R. G., Dawid, A. P., Lauritzen, S. L. & Spiegelhalter, D. J. (1999). Probabilistic Networks and Expert Systems. New York: Springer-Verlag.Google Scholar
Cox, D. R. & Wermuth, N. (1993). Linear dependencies represented by chain graphs. Statistical Science, 8(3), 204–18.Google Scholar
Cox, D. R. & Wermuth, N. (1996). Multivariate Dependencies: Models, Analysis, and Interpretation. Boca Raton, FL: Chapman & Hall.Google Scholar
Darroch, J. N. & Ratcliff, D. (1972). Generalized iterative scaling for log-linear models. The Annals of Mathematical Statistics, 43(5), 1470–80.Google Scholar
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.Google Scholar
Davison, A. C. (2003). Statistical Models. Vol. 11. Cambridge: Cambridge University Press.Google Scholar
Dawid, A. P. (1979). Conditional independence in statistical theory. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 41, 1–31.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Diestel, R. (1990). Graph Decompositions: A Study in Infinite Graph Theory. Oxford: Clarendon Press.Google Scholar
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. 93–104.CrossRefGoogle Scholar
Drton, M. (2009). Discrete chain graph models. Bernoulli, 15(3), 736–53.Google Scholar
Drton, M. & Maathuis, M. H. (2017). Structure learning in graphical modeling. Annual Review of Statistics and Its Application, 4(1).Google Scholar
Drton, M. & Richardson, T. S. (2008a). Binary models for marginal independence. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(2), 287–309.CrossRefGoogle Scholar
Drton, M. & Richardson, T. S. (2008b). Graphical methods for efficient likelihood inference in Gaussian covariance models. Journal of Machine Learning Research, 9, 893–914.Google Scholar
Drton, M., Lauritzen, S. L., Maathuis, M. & Wainwright, M. (2017). Handbook of Graphical Models. Boca Raton, FL: Chapman and Hall/CRC.Google Scholar
Edwards, D. (2000). Introduction to Graphical Modelling, 2nd edn, New York: Springer-Verlag.CrossRefGoogle Scholar
Edwards, D. & Kreiner, S. (1983). The analysis of contingency tables by graphical models. Biometrika, 70(3), 553–65.CrossRefGoogle Scholar
Evans, R. J. (2016). Graphs for margins of Bayesian networks. Scandinavian Journal of Statistics, 43(3), 625–48.Google Scholar
Evans, R. J. & Forcina, A. (2013). Two algorithms for fitting constrained marginal models. Computational Statistics & Data analysis, 66, 1–7.Google Scholar
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.Google Scholar
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.Google Scholar
Evans, R. J. & Richardson, T. S. (2014). Markovian acyclic directed mixed graphs for discrete data. The Annals of Statistics, 42(4), 1452–82.Google Scholar
Frydenberg, M. (1990). The chain graph Markov property. Scandinavian Journal of Statistics, 17(4), 333–353.Google Scholar
Frydenberg, M. & Lauritzen, S. L. (1989). Decomposition of maximum likelihood in mixed graphical interaction models. Biometrika, 76(3), 539–55.Google Scholar
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.Google Scholar
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. 3–14.Google Scholar
Geiger, D. & Pearl, J. (1993). Logical and algorithmic properties of conditional independence and graphical models. The Annals of Statistics, 21(4), 2001–21.Google Scholar
Graybill, F. A. (1983). Matrices with Applications in Statistics. Belmont, CA: Wadsworth.Google Scholar
Gutiérrez-Peña, E. & Smith, A. F. M. (1997). Exponential and Bayesian conjugate families: review and extensions. Test, 6(1), 1–90.Google Scholar
Hall, P. (1934). A contribution to the theory of groups of prime-power order. Proceedings of the London Mathematical Society, 2(1), 29–95.Google Scholar
Hammersley, J. M. & Clifford, P. (1971). Markov fields on finite graphs and lattices. Unpublished manuscript.Google Scholar
Højsgaard, S. (2004). Statistical inference in context specific interaction models for contingency tables. Scandinavian Journal of Statistics, 31(1), 143–58.Google Scholar
Højsgaard, S., Edwards, D. & Lauritzen, S. L. (2012). Graphical Models with R. New York: Springer Science+Business Media.Google Scholar
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.Google ScholarPubMed
Jeffreys, H. (1961). Theory of Probability, 3rd edn. Oxford Classic Texts in the Physical Sciences. Oxford: Oxford University Press.Google Scholar
Jokinen, J. (2006). Fast estimation algorithm for likelihood-based analysis of repeated categorical responses. Computational Statistics & Data Analysis, 51(3), 1509–22.CrossRefGoogle Scholar
Kass, R. E. & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–95.CrossRefGoogle Scholar
Kauermann, G. (1996). On a dualization of graphical Gaussian models. Scandinavian Journal of Statistics, 23(1), 105–16.Google Scholar
Kauermann, G. (1997). A note on multivariate logistic models for contingency tables. Australian Journal of Statistics, 39(3), 261–76.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Lang, J. B. (1996. Maximum likelihood methods for a generalized class of log-linear models. The Annals of Statistics, 24(2), 726–52.CrossRefGoogle Scholar
Lauritzen, S. L. (1996). Graphical models. Oxford: Clarendon Press.Google Scholar
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. 63–107.Google Scholar
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.Google Scholar
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), 31–57.Google Scholar
Lauritzen, S. L., Dawid, A. P., Larsen, B. N. & Leimer, H.-G. (1990). Independence properties of directed Markov fields. Networks, 20(5), 491–505.Google Scholar
Lovász, L. (1993). Combinatorial Problems and Exercises, 2nd edn. Amsterdam: North-Holland.Google Scholar
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), 227–252.Google Scholar
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.CrossRefGoogle Scholar
Lütkepol, H. (1996). Handbook of Matrices. Chichester: Wiley.Google Scholar
Madsen, M. (1976). Statistical analysis of multiple contingency tables. Two examples. Scandinavian Journal of Statistics, 3(3), 97–106.Google Scholar
Marchetti, G. M. & Lupparelli, M. (2011). Chain graph models of multivariate regression type for categorical data. Bernoulli, 17(3), 827–44.CrossRefGoogle Scholar
Massam, H., Liu, J. & Dobra, A. (2009). A conjugate prior for discrete hierarchical log-linear models. The Annals of Statistics, 37(6), 3431–67.Google Scholar
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.Google Scholar
Morris, C. N. (1982). Natural exponential families with quadratic variance functions. The Annals of Statistics, 10(1), 65–80.Google Scholar
Morris, C. N. (1983). Natural exponential families with quadratic variance functions: statistical theory. The Annals of Statistics, 11(2), 515–29.Google Scholar
Nyman, H., Pensar, J., Koski, T. & Corander, J. (2014). Stratified graphical models – context-specific independence in graphical models. Bayesian Analysis, 9(4), 883–908.Google Scholar
Pearl, J. (1986). Fusion, propagation, and structuring in belief networks. Artificial Intelligence, 29(3), 241–88.Google Scholar
Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufmann.Google Scholar
Pearl, J. (2009). Causality, 2nd edn, Cambridge: Cambridge University Press.Google Scholar
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.Google Scholar
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.Google Scholar
Piccioni, M. (2000). Independence structure of natural conjugate densities to exponential families and the Gibbs’ sampler. Scandinavian Journal of Statistics, 27(1), 111–27.Google Scholar
R Core Team. (2016). R: A Language and Environment for Statistical Computing. Vienna: Foundation for Statistical Computing.Google Scholar
Richardson, T. & Spirtes, P. (2002). Ancestral graph Markov models. The Annals of Statistics, 30(4), 962–1030.Google Scholar
Richardson, T. S. (2003). Markov properties for acyclic directed mixed graphs. Scandinavian Journal of Statistics, 30(1), 145–57.CrossRefGoogle Scholar
Rota, G.-C. (1964). On the foundations of combinatorial theory I. Theory of Möbius functions. Probability Theory and Related Fields, 2(4), 340–68.Google Scholar
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), 295–312.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Roverato, A. & La Rocca, L. (2006). On block ordering of variables in graphical modelling. Scandinavian Journal of Statistics, 33(1), 65–81.Google Scholar
Roverato, A. & Studenỳ, M. (2006). A graphical representation of equivalence classes of AMP chain graphs. Journal of Machine Learning Research, 7, 1045–78.Google Scholar
Roverato, A. & Whittaker, J. (1998). The Isserlis matrix and its application to non-decomposable graphical Gaussian models. Biometrika, 85(3), 711–25.CrossRefGoogle Scholar
Roverato, A., Lupparelli, M. & La Rocca, L. (2013). Log-mean linear models for binary data. Biometrika, 100(2), 485–94.Google Scholar
Rudas, T., Bergsma, W. P. & Németh, R. (2010). Marginal log-linear parameterization of conditional independence models. Biometrika, 97(4), 1006–12.Google Scholar
Sadeghi, K. & Lauritzen, S. L. (2014). Markov properties for mixed graphs. Bernoulli, 20(2), 676–96.Google Scholar
Sadeghi, K. & Wermuth, N. (2016). Pairwise Markov properties for regression graphs. Stat, 5, 286–94.Google Scholar
Speed, T. P. (1983). Cumulants and partition lattices. Australian Journal of Statistics, 25(2), 378–88.Google Scholar
Spirtes, P., Glymour, C. & Scheines, R. (2000). Causation, Prediction, and Search, 2nd edn, Cambridge, MA: MIT Press.Google Scholar
Studenỳ, M. (2005). Probabilistic Conditional Independence Structures. London: Springer-Verlag.Google Scholar
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.Google Scholar
Volf, M. & Studenỳ, M. (1999). A graphical characterization of the largest chain graphs. International Journal of Approximate Reasoning, 20(3), 209–36.Google Scholar
Weisner, L. (1935). Abstract theory of inversion of finite series. Transactions of the American Mathematical Society, 38(3), 474–84.CrossRefGoogle Scholar
Wermuth, N. (1976). Model search among multiplicative models. Biometrics, 32(2), 253–63.Google Scholar
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.Google Scholar
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), 21–50.Google Scholar
Wermuth, N. & Sadeghi, K. (2012). Sequences of regressions and their independences. TEST, 21(2), 215–52.Google Scholar
Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Chichester: Wiley.Google Scholar
Wright, S. (1921). Correlation and causation. Journal of Agricultural Research, 20(7), 557–85.Google Scholar

Save element to Kindle

To save this element to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Graphical Models for Categorical Data
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Graphical Models for Categorical Data
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Graphical Models for Categorical Data
Available formats
×