a.k.a Markov random fields, Markov random networks. (other types?)

I would like to know about spatial Poisson random fields, Markov random fields, Bernoulli random fields, especially for discrete multivariate sequences. Gibbs and Boltzman distribution inference.

Wasserman’s explanation of the use case is good: Estimating Undirected Graphs Under Weak Assumptions

## References

Altun, Yasemin, Alex J. Smola, and Thomas Hofmann. 2004. “Exponential Families for Conditional Random Fields.” In

*Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence*, 2–9. UAI ’04. Arlington, Virginia, United States: AUAI Press.Baddeley, A. J., and Marie-Colette NM Van Lieshout. 1995. “Area-Interaction Point Processes.”

*Annals of the Institute of Statistical Mathematics*47 (4): 601–19.Baddeley, A. J., Marie-Colette NM Van Lieshout, and J. Møller. 1996. “Markov Properties of Cluster Processes.”

*Advances in Applied Probability*28 (2): 346–55.Baddeley, Adrian J, Jesper Møller, and Rasmus Plenge Waagepetersen. 2000. “Non- and Semi-Parametric Estimation of Interaction in Inhomogeneous Point Patterns.”

*Statistica Neerlandica*54 (3): 329–50.Baddeley, Adrian, and Jesper Møller. 1989. “Nearest-Neighbour Markov Point Processes and Random Sets.”

*International Statistical Review / Revue Internationale de Statistique*57 (2): 89–121.Bartolucci, Francesco, and Julian Besag. 2002. “A Recursive Algorithm for Markov Random Fields.”

*Biometrika*89 (3): 724–30.Besag, Julian. 1974. “Spatial Interaction and the Statistical Analysis of Lattice Systems.”

*Journal of the Royal Statistical Society. Series B (Methodological)*36 (2): 192–236.———. 1975. “Statistical Analysis of Non-Lattice Data.”

*Journal of the Royal Statistical Society. Series D (The Statistician)*24 (3): 179–95.———. 1986. “On the Statistical Analysis of Dirty Pictures.”

*Journal of the Royal Statistical Society. Series B (Methodological)*48 (3): 259–302.Blake, Andrew, Pushmeet Kohli, and Carsten Rother, eds. 2011.

*Markov Random Fields for Vision and Image Processing*. Cambridge, Mass: MIT Press.Boyd, Stephen. 2010.

*Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers*. Vol. 3. Now Publishers Inc.Bu, Yunqi, and Johannes Lederer. 2017. “Integrating Additional Knowledge Into Estimation of Graphical Models.”

*arXiv:1704.02739 [Stat]*, April.Celeux, Gilles, Florence Forbes, and Nathalie Peyrard. 2003. “EM Procedures Using Mean Field-Like Approximations for Markov Model-Based Image Segmentation.”

*Pattern Recognition*36 (1): 131–44.Cevher, Volkan, Marco F. Duarte, Chinmay Hegde, and Richard Baraniuk. 2009. “Sparse Signal Recovery Using Markov Random Fields.” In

*Advances in Neural Information Processing Systems*, 257–64. Curran Associates, Inc.Clifford, P. 1990. “Markov random fields in statistics.” In

*Disorder in Physical Systems: A Volume in Honour of John Hammersley*, edited by G. R. Grimmett and D. J. A. Welsh. Oxford England : New York: Oxford University Press.Crisan, Dan, and Joaquín Míguez. 2014. “Particle-Kernel Estimation of the Filter Density in State-Space Models.”

*Bernoulli*20 (4): 1879–929.Fallat, Shaun, Steffen Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, and Piotr Zwiernik. 2017. “Total Positivity in Markov Structures.”

*The Annals of Statistics*45 (3): 1152–84.Forbes, F., and N. Peyrard. 2003. “Hidden Markov Random Field Model Selection Criteria Based on Mean Field-Like Approximations.”

*IEEE Transactions on Pattern Analysis and Machine Intelligence*25 (9): 1089–1101.Fridman, Arthur. 2003. “Mixed Markov Models.”

*Proceedings of the National Academy of Sciences*100 (14): 8092–96.Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. 2008. “Sparse Inverse Covariance Estimation with the Graphical Lasso.”

*Biostatistics*9 (3): 432–41.Friel, Nial, and Håvard Rue. 2007. “Recursive Computing and Simulation-Free Inference for General Factorizable Models.”

*Biometrika*94 (3): 661–72.Geyer, Charles J. 1991. “Markov Chain Monte Carlo Maximum Likelihood.”

Geyer, Charles J., and Jesper Møller. 1994. “Simulation Procedures and Likelihood Inference for Spatial Point Processes.”

*Scandinavian Journal of Statistics*, 359–73.Gogate, Vibhav, William Webb, and Pedro Domingos. 2010. “Learning Efficient Markov Networks.” In

*Advances in Neural Information Processing Systems*, 748–56.Goldberg, David A. 2013. “Higher Order Markov Random Fields for Independent Sets.”

*arXiv:1301.1762 [Math-Ph]*, January.Grenander, Ulf. 1989. “Advances in Pattern Theory.”

*The Annals of Statistics*17 (1): 1–30.Griffeath, David. 1976. “Introduction to Random Fields.” In

*Denumerable Markov Chains*, 425–58. Graduate Texts in Mathematics 40. Springer New York.Häggström, Olle, Marie-Colette N. M. van Lieshout, and Jesper Møller. 1999. “Characterization Results and Markov Chain Monte Carlo Algorithms Including Exact Simulation for Some Spatial Point Processes.”

*Bernoulli*5 (4): 641–58.Heckerman, David, David Maxwell Chickering, Christopher Meek, Robert Rounthwaite, and Carl Kadie. 2000. “Dependency Networks for Inference, Collaborative Filtering, and Data Visualization.”

*Journal of Machine Learning Research*1 (Oct): 49–75.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.Jensen, Jens Ledet, and Jesper Møller. 1991. “Pseudolikelihood for Exponential Family Models of Spatial Point Processes.”

*The Annals of Applied Probability*1 (3): 445–61.Jordan, Michael I. 2004. “Graphical Models.”

*Statistical Science*19 (1): 140–55.Jordan, Michael I., Zoubin Ghahramani, Tommi S. Jaakkola, and Lawrence K. Saul. 1999. “An Introduction to Variational Methods for Graphical Models.”

*Machine Learning*37 (2): 183–233.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.———. 2002b. “Probabilistic Inference in Graphical Models.”

*Handbook of Neural Networks and Brain Theory*.Kamenetsky, Dmitri. 2010. “Ising Graphical Model.”

Karlin, Samuel, and Yosef Rinott. 1980. “Classes of Orderings of Measures and Related Correlation Inequalities. I. Multivariate Totally Positive Distributions.”

*Journal of Multivariate Analysis*10 (4): 467–98.Khoshgnauz, Ehsan. 2012. “Learning Markov Network Structure Using Brownian Distance Covariance.”

*arXiv:1206.6361 [Cs, Stat]*, June.Kindermann, Ross P., and J. Laurie Snell. 1980. “On the Relation Between Markov Random Fields and Social Networks.”

*The Journal of Mathematical Sociology*7 (1): 1–13.Kindermann, Ross, and J. Laurie Snell. 1980.

*Markov Random Fields and Their Applications*. Vol. 1. Contemporary Mathematics. Providence, Rhode Island: American Mathematical Society.Krämer, Nicole, Juliane Schäfer, and Anne-Laure Boulesteix. 2009. “Regularized Estimation of Large-Scale Gene Association Networks Using Graphical Gaussian Models.”

*BMC Bioinformatics*10 (1): 384.Krause, Andreas, and Carlos Guestrin. 2009. “Optimal Value of Information in Graphical Models.”

*J. Artif. Int. Res.*35 (1): 557–91.Kschischang, Frank R, Brendan J Frey, and Hans-Andrea Loeliger. 2001. “Factor Graphs and the Sum-Product Algorithm.”

*IEEE Transactions on Information Theory*47 (2): 498–519.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.Lauritzen, Steffen L. 1996.

*Graphical Models*. Oxford Statistical Science Series. Clarendon Press.Lavrenko, Victor, and Jeremy Pickens. 2003a. “Music Modeling with Random Fields.” In

*Proceedings of the 26th Annual International ACM SIGIR Conference on Research and Development in Informaion Retrieval*, 389. ACM Press.———. 2003b. “Polyphonic Music Modeling with Random Fields.” In

*Proceedings of the Eleventh ACM International Conference on Multimedia*, 120. ACM Press.Lederer, Johannes. 2016. “Graphical Models for Discrete and Continuous Data.”

*arXiv:1609.05551 [Math, Stat]*, September.Lee, Su-In, Varun Ganapathi, and Daphne Koller. 2006. “Efficient Structure Learning of Markov Networks Using $ L_1 $-Regularization.” In

*Advances in Neural Information Processing Systems*, 817–24. MIT Press.Liu, Han, Fang Han, Ming Yuan, John Lafferty, and Larry Wasserman. 2012a. “The Nonparanormal SKEPTIC.”

*arXiv:1206.6488 [Cs, Stat]*, June.———. 2012b. “High-Dimensional Semiparametric Gaussian Copula Graphical Models.”

*The Annals of Statistics*40 (4): 2293–2326.Liu, Han, John Lafferty, and Larry Wasserman. 2009. “The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs.”

*Journal of Machine Learning Research*10 (December): 2295–2328.Liu, Han, Kathryn Roeder, and Larry Wasserman. 2010. “Stability Approach to Regularization Selection (StARS) for High Dimensional Graphical Models.” In

*Advances in Neural Information Processing Systems 23*, edited by J. D. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel, and A. Culotta, 1432–40. Curran Associates, Inc.Loeliger, H.-A. 2004. “An Introduction to Factor Graphs.”

*IEEE Signal Processing Magazine*21 (1): 28–41.Maddage, Namunu C., Haizhou Li, and Mohan S. Kankanhalli. 2006. “Music Structure Based Vector Space Retrieval.” In

*Proceedings of the 29th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval*, 67. ACM Press.Malioutov, Dmitry M., Jason K. Johnson, and Alan S. Willsky. 2006. “Walk-Sums and Belief Propagation in Gaussian Graphical Models.”

*Journal of Machine Learning Research*7 (October): 2031–64.Mazumder, Rahul, and Trevor Hastie. 2012. “The graphical lasso: New insights and alternatives.”

*Electronic Journal of Statistics*6 (November): 2125–49.McCallum, Andrew. 2012. “Efficiently Inducing Features of Conditional Random Fields.”

*arXiv:1212.2504 [Cs, Stat]*, October.Meinshausen, Nicolai, and Peter Bühlmann. 2006. “High-Dimensional Graphs and Variable Selection with the Lasso.”

*The Annals of Statistics*34 (3): 1436–62.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.Montanari, Andrea. 2011. “Lecture Notes for Stat 375 Inference in Graphical Models.”

Morgan, Jonathan Scott, Iman Barjasteh, Cliff Lampe, and Hayder Radha. 2014. “The Entropy of Attention and Popularity in Youtube Videos.”

*arXiv:1412.1185 [Physics]*, December.Murphy, Kevin P. 2012a.

*Machine learning: a probabilistic perspective*. 1 edition. Adaptive computation and machine learning series. Cambridge, MA: MIT Press.———. 2012b. “Undirected graphical models (Markov random fields).” In

*Machine Learning: A Probabilistic Perspective*, 1 edition. Cambridge, MA: The MIT Press.Osokin, A., D. Vetrov, and V. Kolmogorov. 2011. “Submodular Decomposition Framework for Inference in Associative Markov Networks with Global Constraints.” In

*2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, 1889–96.Pickens, Jeremy, and Costas S. Iliopoulos. 2005. “Markov Random Fields and Maximum Entropy Modeling for Music Information Retrieval.” In

*ISMIR*, 207–14. Citeseer.Pollard, Dave. 2004. “Hammersley-Clifford Theorem for Markov Random Fields.”

Ranzato, M. 2013. “Modeling Natural Images Using Gated MRFs.”

*IEEE Transactions on Pattern Analysis and Machine Intelligence*35 (9): 2206–22.Ravikumar, Pradeep, Martin J. Wainwright, and John D. Lafferty. 2010. “High-Dimensional Ising Model Selection Using ℓ1-Regularized Logistic Regression.”

*The Annals of Statistics*38 (3): 1287–1319.Reeves, R., and A. N. Pettitt. 2004. “Efficient Recursions for General Factorisable Models.”

*Biometrika*91 (3): 751–57.Richardson, Matthew, and Pedro Domingos. 2006. “Markov Logic Networks.”

*Machine Learning*62 (1-2): 107–36.Ripley, B. D., and F. P. Kelly. 1977. “Markov Point Processes.”

*Journal of the London Mathematical Society*s2-15 (1): 188–92.Schmidt, Mark W., and Kevin P. Murphy. 2010. “Convex Structure Learning in Log-Linear Models: Beyond Pairwise Potentials.” In

*International Conference on Artificial Intelligence and Statistics*, 709–16.Slawski, Martin, and Matthias Hein. 2015. “Estimation of Positive Definite M-Matrices and Structure Learning for Attractive Gaussian Markov Random Fields.”

*Linear Algebra and Its Applications*, Special issue on Statistics, 473 (May): 145–79.Studený, Milan. 1997. “A Recovery Algorithm for Chain Graphs.”

*International Journal of Approximate Reasoning*, Uncertainty in AI (UAI’96) Conference, 17 (2–3): 265–93.———. 2005.

*Probabilistic Conditional Independence Structures*. Information Science and Statistics. London: Springer.Sutton, Charles, and Andrew McCallum. 2010. “An Introduction to Conditional Random Fields.”

*arXiv:1011.4088*, November.Tansey, Wesley, Oscar Hernan Madrid Padilla, Arun Sai Suggala, and Pradeep Ravikumar. 2015. “Vector-Space Markov Random Fields via Exponential Families.” In

*Journal of Machine Learning Research*, 684–92.Vetrov, Dmitry, and Anton Osokin. 2011. “Graph Preserving Label Decomposition in Discrete MRFs with Selfish Potentials.” In

*NIPS Workshop on Discrete Optimization in Machine Learning (DISCML NIPS)*.Visweswaran, Shyam, and Gregory F. Cooper. 2014. “Counting Markov Blanket Structures.”

*arXiv:1407.2483 [Cs, Stat]*, July.Wainwright, Martin J., and Michael I. Jordan. 2008.

*Graphical Models, Exponential Families, and Variational Inference*. Vol. 1. Foundations and Trends® in Machine Learning. Now Publishers.Wainwright, Martin, and Michael I Jordan. 2005. “A Variational Principle for Graphical Models.” In

*New Directions in Statistical Signal Processing*. Vol. 155. MIT Press.Wang, Chaohui, Nikos Komodakis, and Nikos Paragios. 2013. “Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey.”

*Computer Vision and Image Understanding*117 (11): 1610–27.Wasserman, Larry, Mladen Kolar, and Alessandro Rinaldo. 2013. “Estimating Undirected Graphs Under Weak Assumptions.”

*arXiv:1309.6933 [Cs, Math, Stat]*, September.Wu, Rui, R. Srikant, and Jian Ni. 2012. “Learning Graph Structures in Discrete Markov Random Fields.” In

*INFOCOM Workshops*, 214–19.———. 2013. “Learning Loosely Connected Markov Random Fields.”

*Stochastic Systems*3 (2): 362–404.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.———. 2005. “Constructing Free-Energy Approximations and Generalized Belief Propagation Algorithms.”

*IEEE Transactions on Information Theory*51 (7): 2282–312.Zhao, Tuo, Han Liu, Kathryn Roeder, John Lafferty, and Larry Wasserman. 2012. “The Huge Package for High-Dimensional Undirected Graph Estimation in R.”

*Journal of Machine Learning Research : JMLR*13 (April): 1059–62.Zhu, Wanchuang, and Yanan Fan. 2022. “A Synthetic Likelihood Approach for Intractable Markov Random Fields.”

*Computational Statistics*, July.
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