{ "id": "eprv797860", "slug": "graphical-model", "title": "Graphical model", "updated": "2026-10-25", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis" }, { "id": "physical.math.statistics.multivariate-analysis.alpha-a-to-me", "label": "Titles A to Me", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis.alpha-a-to-me" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "label": "Non-geographic · 1946 to 2000: Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis" } ] } ], "excerpt": "A graphical model represents a joint probability distribution as a graph, with nodes for variables and edges for direct interactions; missing arcs encode conditional independence.", "snippet": "A graphical model represents a joint probability distribution as a graph, with nodes for variables and edges for direct interactions; missing arcs encode conditional independence.", "node": "physical.math.statistics.multivariate-analysis.alpha-a-to-me", "markdown": "# Graphical model\n\nA graphical model represents a [joint probability distribution](https://www.edgepedia.org/joint-probability-distribution) over random variables as a graph, with nodes standing for the variables and edges for direct probabilistic interactions between them. The (lack of) arcs encodes conditional independence assumptions, so the graph is a compact representation of a high-dimensional distribution rather than a picture of it.[1](https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4347-Koller_Friedman%20%282009%29%20-%20Probabilistic%20graphical%20models.pdf) • [2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf) Every model has a structural component, the pattern of edges, and a parametric component, numerical factors attached to conditional distributions or to cliques of variables.[3](https://mlg.eng.cam.ac.uk/zoubin/course03/hbtnn2e-I.pdf) Two families dominate: Bayesian networks, which use directed acyclic graphs, and Markov networks, also called Markov random fields, which use undirected graphs.[1](https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4347-Koller_Friedman%20%282009%29%20-%20Probabilistic%20graphical%20models.pdf) • [4](https://www.deeplearningbook.org/contents/graphical_models.html)\n\n| Key fact | Statement | Source |\n|---|---|---|\n| Representation | Nodes are random variables; edges are direct probabilistic interactions | [1](https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4347-Koller_Friedman%20%282009%29%20-%20Probabilistic%20graphical%20models.pdf) |\n| Missing arcs | Encode conditional independence assumptions | [2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf) |\n| Factorization gain | An atomic joint over \\( N \\) binary variables needs \\( O(2^{N}) \\) parameters; a factored network with \\( n \\) binary nodes and maximum fan-in \\( k \\) needs \\( O(n \\cdot 2^{k}) \\) | [2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf) |\n| d-separation | Necessary and sufficient for the distributions a DAG represents; testable in time linear in the number of edges | [5](https://ftp.cs.ucla.edu/pub/stat_ser/r236-3ed.pdf) |\n| Exact inference cost | Exponential in the largest clique; treewidth is 1 for a chain, while the treewidth of a two-dimensional grid grows with the shorter side length and is unbounded as the grid grows | [6](http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf) |\n| Hardness | Every type of inference is NP-hard or harder; computing the partition function is #P-complete | [6](http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf) • [7](https://drops.dagstuhl.de/storage/00lipics/lipics-vol154-stacs2020/LIPIcs.STACS.2020.4/LIPIcs.STACS.2020.4.pdf) |\n| Inference classes | Exact algorithms, sampling algorithms, and variational algorithms | [8](https://oa.ee.tsinghua.edu.cn/~ouzhijian/pgm/pgm-pdf/GraphicalModels2004.pdf) |\n\n## How it works\n\n**Directed factorization.** A Bayesian network is a directed acyclic graph whose joint distribution is the product of one conditional distribution per node [8](https://oa.ee.tsinghua.edu.cn/~ouzhijian/pgm/pgm-pdf/GraphicalModels2004.pdf):\n\n\\[ p(X_{1},\\ldots,X_{n}) = \\prod_{i=1}^{n} p(X_{i} \\mid X_{\\mathrm{pa}(i)}) \\]\n\nwhere \\( \\mathrm{pa}(i) \\) are the parents of node \\( i \\).[9](https://mlg.eng.cam.ac.uk/zoubin/talks/lect2gm.pdf) The graph's local property is that each variable is independent of its non-descendants given its parents.[10](https://ftp.cs.ucla.edu/tech-report/198_-reports/850017.pdf) [Factorization](https://www.edgepedia.org/factorization) replaces exponential growth in the number of variables with exponential growth in the number of parent configurations.[2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf)\n\n**d-separation.** d-separation reads conditional independence directly from the graph. A set \\( V \\) d-separates \\( X \\) from \\( Y \\) if every undirected path between them is blocked.[11](https://bayes.cs.ucla.edu/PRIMER/primer-ch2.pdf) With no conditioning, only colliders (converging arrows) can block a path; conditioning on a collider or one of its descendants unblocks it.[11](https://bayes.cs.ucla.edu/PRIMER/primer-ch2.pdf) The criterion is sound and complete for the conditional-independence implications of the DAG, and it can be tested in time linear in the number of edges; for a particular distribution, the converse requires a faithfulness assumption.[5](https://ftp.cs.ucla.edu/pub/stat_ser/r236-3ed.pdf)\n\n**Undirected models.** In a [Markov random field](https://www.edgepedia.org/markov-random-field), \\( X_{A} \\) is independent of \\( X_{B} \\) given \\( X_{C} \\) when removing the vertices \\( C \\) leaves no path from \\( A \\) to \\( B \\).[12](https://www.cs.columbia.edu/~blei/fogm/2023F/readings/WainwrightJordan2008.pdf) The Hammersley-Clifford theorem states that a positive distribution has the independence structure of the graph if and only if it factorizes over the maximal cliques, \\( p(x_{1},\\ldots,x_{N}) = Z^{-1} \\prod_{c} \\Psi_{c}(x_{c}) \\), where \\( Z \\) is the partition function [13](https://visionbook.mit.edu/graphical_models.html); such a product is called a Gibbs distribution, with a normalizing partition function.[6](http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf) Converting a directed graph to its moral graph, by marrying parents, makes the directed factorization a special case of the undirected one.[8](https://oa.ee.tsinghua.edu.cn/~ouzhijian/pgm/pgm-pdf/GraphicalModels2004.pdf)\n\n## How it is done\n\n**Exact inference.** [Variable elimination](https://www.edgepedia.org/variable-elimination) works by pushing sums in as far as possible.[2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf) [Message passing](https://www.edgepedia.org/message-passing) on trees and polytrees computes a conditional probability such as \\( p(x_{1} \\mid y) \\) in \\( O(n \\cdot N^{2}) \\) operations for \\( N \\)-valued discrete variables, against \\( O(N^{n-1}) \\) for naive integration.[14](https://pages.cs.wisc.edu/~jerryzhu/cs761/graphical_model_note.pdf) Clique-tree propagation, the basis of the junction tree method, converts a graph with loops into a tree of clusters and runs message passing on it [15](https://doi.org/10.1111/j.2517-6161.1988.tb01721.x) • [2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf); in a junction tree, local consistency implies global consistency, so the local algorithm is provably correct.[16](https://www.cis.upenn.edu/~mkearns/papers/barbados/jordan-tut.pdf) The max-product version of belief propagation computes MAP estimates, and applied to hidden Markov models it is the [Viterbi algorithm](https://www.edgepedia.org/viterbi-algorithm).[17](https://www.stat.cmu.edu/%7Elarry/=sml/DAGs.pdf)\n\n**Approximate inference.** The three principal classes are exact algorithms, sampling algorithms, and variational algorithms.[8](https://oa.ee.tsinghua.edu.cn/~ouzhijian/pgm/pgm-pdf/GraphicalModels2004.pdf) [Gibbs sampling](https://www.edgepedia.org/gibbs-sampling) conditions each update only on the [Markov blanket](https://www.edgepedia.org/markov-blanket), which in a directed graph is the set of parents, children, and co-parents.[16](https://www.cis.upenn.edu/~mkearns/papers/barbados/jordan-tut.pdf) Variational methods are deterministic approximations that tend to work best for dense graphs and yield upper or lower bounds on probabilities.[16](https://www.cis.upenn.edu/~mkearns/papers/barbados/jordan-tut.pdf) Loopy belief propagation applies tree message updates to graphs with cycles; its convergence is not guaranteed [17](https://www.stat.cmu.edu/%7Elarry/=sml/DAGs.pdf), and an empirical study by Kevin Murphy, Yair Weiss, and [Michael I. Jordan](https://www.edgepedia.org/michael-i-jordan) examined its behavior on such networks.[18](https://doi.org/10.48550/arxiv.1301.6725)\n\n**Learning.** [Structure](https://www.edgepedia.org/structure) learning follows two routes: constraint-based, using statistical tests of marginal and conditional independence to find matching d-separation relations, and score-based, using a global score such as BIC or the Bayesian marginal likelihood with greedy search or MCMC.[9](https://mlg.eng.cam.ac.uk/zoubin/talks/lect2gm.pdf) Identifying a high-scoring structure is NP-hard even with independence, inference, or information oracles, and even when each node is limited to at most \\( k \\) parents for all \\( k \\geq 3 \\).[19](https://jmlr.org/papers/volume5/chickering04a/chickering04a.pdf) For Gaussian models, the zero entries of the inverse covariance (precision) matrix determine the edges [14](https://pages.cs.wisc.edu/~jerryzhu/cs761/graphical_model_note.pdf), and the graphical lasso of Jerome Friedman, Trevor Hastie, and [Robert Tibshirani](https://www.edgepedia.org/robert-tibshirani) (2007) estimates a sparse precision matrix.[20](https://doi.org/10.1093/biostatistics/kxm045) Because treewidth, not the number of variables, dictates the cost of exact inference, it is worth checking several tree decompositions before resorting to approximation.[21](https://onlinelibrary.wiley.com/doi/10.1111/anzs.12257)\n\n## Origin\n\nDirected-graph representations of inheritance and of path analysis in genetics and social science predate the modern formalism, and the clique-potential naming has roots in statistical physics.[6](http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf) The modern literature consolidated in a cluster of work from the 1980s. Bayesian networks are directed acyclic graphs of propositions whose conditional-independence statements can be tested with simple link-tracing operations [10](https://ftp.cs.ucla.edu/tech-report/198_-reports/850017.pdf), and his 1988 book treatment, published by Elsevier, covers Markov and Bayesian networks together.[22](https://doi.org/10.1016/b978-0-08-051489-5.50009-6) Lauritzen and Spiegelhalter's 1988 paper in the Journal of the Royal Statistical Society Series B on local computations with probabilities introduced clique-tree propagation for expert systems.[15](https://doi.org/10.1111/j.2517-6161.1988.tb01721.x) Geiger and Pearl (1993) analyzed the logical and algorithmic properties linking conditional independence to graph separation in The Annals of Statistics [23](https://doi.org/10.1214/aos/1176349407), and Robertson and Seymour (1986) developed treewidth in their graph-minors series in the Journal of Algorithms.[24](https://doi.org/10.1016/0196-6774%2886%2990023-4) Later unifications include Michael I. Jordan's 2004 survey in Statistical Science [8](https://oa.ee.tsinghua.edu.cn/~ouzhijian/pgm/pgm-pdf/GraphicalModels2004.pdf), the factor-graph formalism of Kschischang, Frey, and Loeliger (2001) in the IEEE Transactions on Information Theory [25](https://doi.org/10.1109/18.910572), and the graphical lasso (2007) in [Biostatistics](https://www.edgepedia.org/biostatistics).[20](https://doi.org/10.1093/biostatistics/kxm045)\n\n## Variants\n\n**Hidden Markov models** are chain-structured models with three fundamental problems: evaluating the likelihood of an observation sequence, determining the best state sequence, and adjusting model parameters to account for the observed signal.[26](https://web.mit.edu/6.435/www/Rabiner89.pdf) The forward-backward algorithm and Kalman smoothing are special cases of belief propagation and of the junction tree algorithm.[9](https://mlg.eng.cam.ac.uk/zoubin/talks/lect2gm.pdf) • [16](https://www.cis.upenn.edu/~mkearns/papers/barbados/jordan-tut.pdf) [Factorial](https://www.edgepedia.org/factorial) hidden Markov models were presented by [Zoubin Ghahramani](https://www.edgepedia.org/zoubin-ghahramani) and Michael I. Jordan (1997) in Machine Learning.[27](https://doi.org/10.1023/a:1007425814087)\n\n**Ising models** place variables on a grid with edge potentials that penalize differences between neighbors, \\( \\Psi(x_{i}, x_{j}) = e^{J(x_{i}, x_{j})} \\), where \\( J(x_{i}, x_{j}) = 1 \\) if \\( x_{i} = x_{j} \\) and \\( -1 \\) otherwise.[2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf)\n\n**Factor graphs** are bipartite graphs with round variable nodes and square factor nodes, representing distributions as products of factors.[25](https://doi.org/10.1109/18.910572) Hybrid models combining directed and undirected components include Deep Belief Networks.[28](https://www.cs.cmu.edu/~rsalakhu/10417/Lectures/Lecture_GM.pdf) On the inference side, the Factor Graph Neural Network of Zhen Zhang, Fan Wu, and Wee Sun Lee (2019) can exactly parameterize the max-product loopy belief propagation algorithm.[29](https://doi.org/10.48550/arxiv.1906.00554)\n\n## Applications\n\nGraphical models have been used for medical and fault diagnosis, for modeling human genetic inheritance of disease, for segmenting and denoising images, for decoding messages sent over a noisy channel, for revealing genetic regulatory networks, and for robot localization and mapping.[1](https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4347-Koller_Friedman%20%282009%29%20-%20Probabilistic%20graphical%20models.pdf) Special cases of Bayes nets were independently invented by many communities: genetics (linkage analysis), speech recognition (HMMs), tracking (Kalman filtering), data compression, and channel coding (turbo codes).[2](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf) In error-control coding, the Bethe approximation underlying loopy message passing has allowed practical codes to nearly reach the Shannon limit.[3](https://mlg.eng.cam.ac.uk/zoubin/course03/hbtnn2e-I.pdf)\n\n## Limitations and alternatives\n\n**Complexity.** Every type of inference in graphical models is NP-hard or harder; even computing the distribution over a single binary variable is NP-hard, and approximating it within \\( \\varepsilon \\in (0, 1/2) \\) is NP-hard.[6](http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf) Computing the partition function is #P-complete, as is model counting.[7](https://drops.dagstuhl.de/storage/00lipics/lipics-vol154-stacs2020/LIPIcs.STACS.2020.4/LIPIcs.STACS.2020.4.pdf) The treewidth of a chordal graph is the size of its largest clique minus 1; a chain has treewidth 1, and the treewidth of a two-dimensional grid grows with its shorter side length and is unbounded as the grid grows.[6](http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf)\n\n**Algorithmic limits.** Loopy belief propagation has no general convergence guarantee.[17](https://www.stat.cmu.edu/%7Elarry/=sml/DAGs.pdf)\n\n**Directed versus undirected, and neural alternatives.** Neither family is clearly superior and universally preferred; some distributions are represented more efficiently by one than the other.[4](https://www.deeplearningbook.org/contents/graphical_models.html) [Ancestral sampling](https://www.edgepedia.org/ancestral-sampling) is efficient but applies only to directed models, while drawing samples from an undirected model is an expensive, multipass process, and the undirected partition function is often intractable to compute.[4](https://www.deeplearningbook.org/contents/graphical_models.html) Probabilistic graphical models represent factorized probability distributions and pass messages to compute probabilities, whereas neural networks compute results that minimize an expected loss or an energy function [13](https://visionbook.mit.edu/graphical_models.html); Boltzmann machines and mixtures of experts are special cases of graphical models.[16](https://www.cis.upenn.edu/~mkearns/papers/barbados/jordan-tut.pdf)\n\n## References\n\n1. [4347 Koller Friedman (2009) Probabilistic graphical models (wcl.cs.rpi.edu)](https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4347-Koller_Friedman%20%282009%29%20-%20Probabilistic%20graphical%20models.pdf)\n2. [An introduction to graphical models (Murphy; merged with identical copy at cs.cmu.edu)](https://www.cs.ubc.ca/~murphyk/papers/intro_gm.pdf)\n3. [Probabilistic inference in graphical models (Jordan & Weiss, chapter draft)](https://mlg.eng.cam.ac.uk/zoubin/course03/hbtnn2e-I.pdf)\n4. [Deep Learning, Chapter 16: Structured Probabilistic Models for Deep Learning (Goodfellow, Bengio, Courville)](https://www.deeplearningbook.org/contents/graphical_models.html)\n5. [Graphical Models for Probabilistic and Causal Reasoning (Pearl, R-236, 3rd ed.; merged with R-236 2nd edition 2011 copy)](https://ftp.cs.ucla.edu/pub/stat_ser/r236-3ed.pdf)\n6. [Graphical Models in a Nutshell (Koller et al., SRL book chapter)](http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf)\n7. [Graphical Models: Queries, Complexity, Algorithms (STACS 2020)](https://drops.dagstuhl.de/storage/00lipics/lipics-vol154-stacs2020/LIPIcs.STACS.2020.4/LIPIcs.STACS.2020.4.pdf)\n8. [Graphical Models (M. I. Jordan, Statistical Science 2004)](https://oa.ee.tsinghua.edu.cn/~ouzhijian/pgm/pgm-pdf/GraphicalModels2004.pdf)\n9. [Graphical Models lecture (Ghahramani, Cambridge)](https://mlg.eng.cam.ac.uk/zoubin/talks/lect2gm.pdf)\n10. [Bayesian networks (Pearl, UCLA technical report CSD-850017, 1985)](https://ftp.cs.ucla.edu/tech-report/198_-reports/850017.pdf)\n11. [Primer on Causal Inference, Chapter 2 (Pearl, Glymour, Jewell)](https://bayes.cs.ucla.edu/PRIMER/primer-ch2.pdf)\n12. [Graphical Models, Exponential Families, and Variational Inference (Wainwright & Jordan, 2008)](https://www.cs.columbia.edu/~blei/fogm/2023F/readings/WainwrightJordan2008.pdf)\n13. [Probabilistic Graphical Models – Foundations of Computer Vision (Freeman et al.)](https://visionbook.mit.edu/graphical_models.html)\n14. [Graphical Models lecture notes (Zhu, UW-Madison)](https://pages.cs.wisc.edu/~jerryzhu/cs761/graphical_model_note.pdf)\n15. [S. L. Lauritzen, 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 (Statistical Methodology).](https://doi.org/10.1111/j.2517-6161.1988.tb01721.x)\n16. [An Introduction to Graphical Models (Jordan, Barbados tutorial)](https://www.cis.upenn.edu/~mkearns/papers/barbados/jordan-tut.pdf)\n17. [Directed Graphical Models (Wasserman, Statistical Machine Learning notes)](https://www.stat.cmu.edu/%7Elarry/=sml/DAGs.pdf)\n18. [Murphy, Kevin, Weiss, Yair, Jordan, Michael I. (2013). Loopy Belief Propagation for Approximate Inference: An Empirical Study. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1301.6725)\n19. [Large-Sample Learning of Bayesian Networks is NP-Hard (Chickering, JMLR)](https://jmlr.org/papers/volume5/chickering04a/chickering04a.pdf)\n20. [Jerome Friedman, Trevor Hastie, Robert Tibshirani (2007). Sparse inverse covariance estimation with the graphical lasso. Biostatistics.](https://doi.org/10.1093/biostatistics/kxm045)\n21. [Exact or approximate inference in graphical models: why the choice is dictated by the treewidth (Australian & New Zealand Journal of Statistics)](https://onlinelibrary.wiley.com/doi/10.1111/anzs.12257)\n22. [Judea Pearl (1988). MARKOV AND BAYESIAN NETWORKS. Elsevier eBooks.](https://doi.org/10.1016/b978-0-08-051489-5.50009-6)\n23. [Dan Geiger, Judea Pearl (1993). Logical and Algorithmic Properties of Conditional Independence and Graphical Models. The Annals of Statistics.](https://doi.org/10.1214/aos/1176349407)\n24. [Graph minors. II. Algorithmic aspects of tree-width (Journal of Algorithms, 1986)](https://doi.org/10.1016/0196-6774%2886%2990023-4)\n25. [F.R. Kschischang, B.J. Frey, H.-A. Loeliger (2001). Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory.](https://doi.org/10.1109/18.910572)\n26. [A tutorial on hidden Markov models and selected applications in speech recognition (Rabiner, Proceedings of the IEEE 1989)](https://web.mit.edu/6.435/www/Rabiner89.pdf)\n27. [Zoubin Ghahramani, Michael I. Jordan (1997). Factorial Hidden Markov Models. Machine Learning.](https://doi.org/10.1023/a:1007425814087)\n28. [Graphical Models I lecture (Salakhutdinov, CMU)](https://www.cs.cmu.edu/~rsalakhu/10417/Lectures/Lecture_GM.pdf)\n29. [Zhang, Zhen, Wu, Fan, Lee, Wee Sun (2019). Factor Graph Neural Network. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1906.00554)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Multivariate association and dimension reduction › Titles A to Me*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 25, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "http://ai.stanford.edu/~koller/Papers/Koller+al%3ASRL07.pdf", "https://www.cs.columbia.edu/~blei/fogm/2023F/readings/WainwrightJordan2008.pdf", "https://pages.cs.wisc.edu/~jerryzhu/cs761/graphical_model_note.pdf", "https://www.cis.upenn.edu/~mkearns/papers/barbados/jordan-tut.pdf", "https://www.cs.cmu.edu/~rsalakhu/10417/Lectures/Lecture_GM.pdf" ], "url": "https://www.edgepedia.org/graphical-model", "markdown_url": "https://www.edgepedia.org/graphical-model.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Graphical model\", Edgepedia (EdgeChat), https://www.edgepedia.org/graphical-model. Edgepedia Community License 1.0.", "credit_md": "\"[Graphical model](https://www.edgepedia.org/graphical-model)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/graphical-model](https://www.edgepedia.org/graphical-model). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Graphical model\", Edgepedia (EdgeChat), https://www.edgepedia.org/graphical-model. Edgepedia Community License 1.0.", "speakable": "A graphical model represents a joint probability distribution as a graph, with nodes for variables and edges for direct interactions; missing arcs encode conditional independence." }