The DOI matches its Crossref record; 2 of the 2 references it cites were found in a registry or an index.
An Introduction to Variational Methods for Graphical Models
Open accessOverview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
A variety of analytic approximations exist which ameliorate, but do not remove, the tradeoff between tractability and flexibility.
“A variety of analytic approximations exist which ameliorate, but do not remove, this tradeoff”
From Deep Unsupervised Learning using Nonequilibrium Thermodynamics (2015) · cites this paper as Jordan et al., 1999
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Maximum Likelihood from Incomplete Data Via the EM Algorithm
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1977.tb01600.x · Cited by 50,444
Keeping the neural networks simple by minimizing the description length of the weights
Hinton, G. E., & van Camp, D. (1993) · Proceedings of the sixth annual conference on Computational learning theory - COLT '93 · doi:10.1145/168304.168306 · Cited by 994
Flagged first, then by citations
Research works printed but not linked: 59
- Bathe, K. J. (1996). Finite element procedures. Englewood Cliffs, NJ: Prentice-Hall.
- Baum, L. E., Petrie, T., Soules, G., & Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. The Annals of Mathematical Statistics, 41, 164–171. · DOI
- Bishop, C. M., Lawrence, N., Jaakkola, T. S., & Jordan, M. I. (1998). Approximating posterior distributions in belief networks using mixtures. In M. Jordan, M. Kearns, & S. Solla (Eds.), Advances in neural information processing systems 10, Cambridge MA: MIT Press.
- Cover, T., & Thomas, J. (1991). Elements of information theory. New York: John Wiley.
- Dagum, P., & Luby, M. (1993). Approximating probabilistic inference in Bayesian belief networks is NP-hard. Artificial Intelligence, 60, 141–153. · DOI
- Dayan, P., Hinton, G. E., Neal, R., & Zemel, R. S. (1995). The Helmholtz Machine. Neural Computation, 7, 889–904. · DOI · Paper page
- Dean, T., & Kanazawa, K. (1989). A model for reasoning about causality and persistence. Computational Intelligence, 5, 142–150. · DOI · Paper page
- Dechter, R. (1999). Bucket elimination: A unifying framework for probabilistic inference. In M. I. Jordan (Ed.), Learning in graphical models. Cambridge, MA: MIT Press.
- Draper, D. L., & Hanks, S. (1994). Localized partial evaluation of belief networks. Uncertainty and Artificial Intelligence: Proceedings of the Tenth Conference. San Mateo, CA: Morgan Kaufmann.
- Frey, B., Hinton, G. E., & Dayan, P. (1996). Does the wake-sleep algorithm learn good density estimators? In D. S. Touretzky, M. C. Mozer, & M. E. Hasselmo (Eds.), Advances in neural information processing systems 8. Cambridge, MA: MIT Press.
- Fung, R. & Favero, B. D. (1994). Backward simulation in Bayesian networks. Uncertainty and Artificial Intelligence: Proceedings of the Tenth Conference. San Mateo, CA: Morgan Kaufmann.
- Galland, C. (1993). The limitations of deterministic Boltzmann machine learning. Network, 4, 355–379. · DOI
- Ghahramani, Z., & Hinton, G. E. (1996). Switching state-space models. (Technical Report CRG-TR–96–3). Toronto: Department of Computer Science, University of Toronto.
- Ghahramani, Z., & Jordan, M. I. (1997). Factorial Hidden Markov models. Machine Learning, 29, 245–273. · DOI · Paper page
- Gilks, W., Thomas, A., & Spiegelhalter, D. (1994). A language and a program for complex Bayesian modelling. The Statistician, 43, 169–178. · DOI
- Heckerman, D. (1999). A tutorial on learning with Bayesian networks. In M. I. Jordan (Ed.), Learning in graphical models. Cambridge, MA: MIT Press.
- Henrion, M. (1991). Search-based methods to bound diagnostic probabilities in very large belief nets. Uncertainty and Artificial Intelligence: Proceedings of the Seventh Conference. San Mateo, CA: Morgan Kaufmann.
- Hinton, G. E., & Sejnowski, T. (1986). Learning and relearning in Boltzmann machines. In D. E. Rumelhart & J. L. McClelland (Eds.), Parallel distributed processing (Vol. 1). Cambridge, MA: MIT Press.
- Hinton, G. E., Dayan, P., Frey, B., & Neal, R. M. (1995). The wake-sleep algorithm for unsupervised neural networks. Science, 268, 1158–1161. · DOI · Paper page
- Hinton, G. E., Sallans, B., & Ghahramani, Z. (1999). A hierarchical community of experts. In M. I. Jordan (Ed.), Learning in graphical models. Cambridge, MA: MIT Press.
- Horvitz, E. J., Suermondt, H. J., & Cooper, G. F. (1989). Bounded conditioning: Flexible inference for decisions under scarce resources. Conference on Uncertainty in Artificial Intelligence: Proceedings of the Fifth Conference. Mountain View, CA: Association for UAI.
- Jaakkola, T. S., & Jordan, M. I. (1996). Computing upper and lower bounds on likelihoods in intractable networks. Uncertainty and Artificial Intelligence: Proceedings of the Twelth Conference. San Mateo, CA: Morgan Kaufmann.
- Jaakkola, T. S. (1997). Variational methods for inference and estimation in graphical models. Unpublished doctoral dissertation, Massachusetts Institute of Technology, Cambridge, MA.
- Jaakkola, T. S., & Jordan, M. I. (1997a). Recursive algorithms for approximating probabilities in graphical models. In M. C. Mozer, M. I. Jordan, & T. Petsche (Eds.), Advances in neural information processing systems 9. Cambridge, MA: MIT Press.
- Jaakkola, T. S., & Jordan, M. I. (1997b). Bayesian logistic regression: A variational approach. In D. Madigan & P. Smyth (Eds.), Proceedings of the 1997 Conference on Artificial Intelligence and Statistics. Ft. Lauderdale, FL.
- Jaakkola, T. S., & Jordan, M. I. (1999a). Improving the mean field approximation via the use of mixture distributions. In M. I. Jordan (Ed.), Learning in graphical models. Cambridge, MA: MIT Press.
- Jaakkola, T. S., & Jordan, M. I. (1999b). Variational methods and the QMR-DT database. Journal of Artificial Intelligence Research, 10, 291–322. · DOI
- Jensen, C. S., Kong, A., & Kjærulff, U. (1995). Blocking-Gibbs sampling in very large probabilistic expert systems. International Journal of Human-Computer Studies, 42, 647–666. · DOI
- Jensen, F. V., & Jensen, F. (1994). Optimal junction trees. Uncertainty and Artificial Intelligence: Proceedings of the Tenth Conference. San Mateo, CA: Morgan Kaufmann.
- Jensen, F. V. (1996). An introduction to Bayesian networks. London: UCL Press.
- Jordan, M. I. (1994). A statistical approach to decision tree modeling. In M. Warmuth (Ed.), Proceedings of the Seventh Annual ACM Conference on Computational Learning Theory. New York: ACM Press.
- Jordan, M. I., Ghahramani, Z., & Saul, L. K. (1997). Hidden Markov decision trees. In M. C. Mozer, M. I. Jordan, & T. Petsche (Eds.), Advances in neural information processing systems 9. Cambridge, MA: MIT Press.
- Kanazawa, K., Koller, D., & Russell, S. (1995). Stochastic simulation algorithms for dynamic probabilistic networks. Uncertainty and Artificial Intelligence: Proceedings of the Eleventh Conference. San Mateo, CA: Morgan Kaufmann.
- Kjærulff, U. (1990). Triangulation of graphs—Algorithms giving small total state space. (Research Report R–90–09). Department of Mathematics and Computer Science, Aalborg University, Denmark.
- Kjærulff, U. (1994). Reduction of computational complexity in Bayesian networks through removal of weak dependences. Uncertainty and Artificial Intelligence: Proceedings of the Tenth Conference. San Mateo, CA: Morgan Kaufmann.
- MacKay, D. J. C. (1997). Ensemble learning for hidden Markov models. Unpublished manuscript. Cambridge: Department of Physics, University of Cambridge.
- McEliece, R. J., MacKay, D. J. C., & Cheng, J.-F. (1998). Turbo decoding as an instance of Pearl's “belief propagation algorithm.” IEEE Journal on Selected Areas in Communication, 16, 140–152. · DOI
- Merz, C. J., & Murphy, P. M. (1996). UCI repository of machine learning databases. Irvine, CA: Department of Information and Computer Science, University of California.
- Neal, R. (1992). Connectionist learning of belief networks. Artificial Intelligence, 56, 71–113. · DOI
- Neal, R. (1993). Probabilistic inference using Markov chain Monte Carlo methods. (Technical Report CRG-TR–93–1). Toronto: Department of Computer Science, University of Toronto.
- Neal, R., & Hinton, G. E. (1999). A view of the EM algorithm that justifies incremental, sparse, and other variants. In M. I. Jordan (Ed.), Learning in graphical models. Cambridge, MA: MIT Press.
- Parisi, G. (1988). Statistical field theory. Redwood City, CA: Addison-Wesley.
- Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Mateo, CA: Morgan Kaufmannn.
- Peterson, C., & Anderson, J. R. (1987). A mean field theory learning algorithm for neural networks. Complex Systems, 1, 995–1019.
- Rockafellar, R. (1972). Convex analysis. Princeton University Press.
- Rustagi, J. (1976). Variational methods in statistics. New York: Academic Press.
- Sakurai, J. (1985). Modern quantum mechanics. Redwood City, CA: Addison-Wesley.
- Saul, L. K., & Jordan, M. I. (1994). Learning in Boltzmann trees. Neural Computation, 6, 1173–1183.
- Saul, L. K., Jaakkola, T. S., & Jordan, M. I. (1996). Mean field theory for sigmoid belief networks. Journal of Artificial Intelligence Research, 4, 61–76. · DOI
- Saul, L. K., & Jordan, M. I. (1996). Exploiting tractable substructures in intractable networks. In D. S. Touretzky, M. C. Mozer, & M. E. Hasselmo (Eds.), Advances in neural information processing systems 8. Cambridge, MA: MIT Press.
- Saul, L. K., & Jordan, M. I. (1999). A mean field learning algorithm for unsupervised neural networks. In M. I. Jordan (Ed.), Learning in graphical models. Cambridge, MA: MIT Press.
- Seung, S. (1995). Annealed theories of learning. In J.-H. Oh, C. Kwon, S. Cho (Eds.), Neural networks: The statistical mechanics perspectives. Singapore: World Scientific.
- Shachter, R. D., Andersen, S. K., & Szolovits, P. (1994). Global conditioning for probabilistic inference in belief networks. Uncertainty and Artificial Intelligence: Proceedings of the Tenth Conference. San Mateo, CA: Morgan Kaufmann.
- Shenoy, P. P. (1992). Valuation-based systems for Bayesian decision analysis. Operations Research, 40, 463–484. · DOI
- Shwe, M. A., & Cooper, G. F. (1991). An empirical analysis of likelihood—Weighting simulation on a large, multiply connected medical belief network. Computers and Biomedical Research, 24, 453–475. · DOI
- Shwe, M. A., Middleton, B., Heckerman, D. E., Henrion, M., Horvitz, E. J., Lehmann, H. P., & Cooper, G. F. (1991). Probabilistic diagnosis using a reformulation of the INTERNIST-1/QMR knowledge base. Meth. Inform. Med., 30, 241–255. · DOI
- Smyth, P., Heckerman, D., & Jordan, M. I. (1997). Probabilistic independence networks for hidden Markov probability models. Neural Computation, 9, 227–270. · DOI
- Waterhouse, S., MacKay, D. J. C., & Robinson, T. (1996). Bayesian methods for mixtures of experts. In D. S. Touretzky, M. C. Mozer, & M. E. Hasselmo (Eds.), Advances in neural information processing systems 8. Cambridge, MA: MIT Press.
- Williams, C. K. I., & Hinton, G. E. (1991). Mean field networks that learn to discriminate temporally distorted strings. In D. S. Touretzky, J. Elman, T. Sejnowski, & G. E. Hinton (Eds.), Proceedings of the 1990 Connectionist Models Summer School. San Mateo, CA: Morgan Kaufmann.
Cited by
Sorted by citations
Fortunato, S. (2010) · Physics Reports · doi:10.1016/j.physrep.2009.11.002 · Cited by 11,293
Collective Classification in Network Data
Sen, P., Namata, G., Bilgic, M., Getoor, L., Gallagher, B., & Eliassi‐Rad, T. (2008) · AI Magazine · doi:10.1609/aimag.v29i3.2157 · Cited by 3,293
A review of uncertainty quantification in deep learning: Techniques, applications and challenges
Abdar, M., Pourpanah, F., Hussain, S., Rezazadegan, D., Liu, L., Ghavamzadeh, M., Fieguth, P., Cao, X., Khosravi, A., Acharya, U. R., et al. (2021) · Information Fusion · doi:10.1016/j.inffus.2021.05.008 · Cited by 3,111
“The minimization objective is as follows [23]:”
Deep Unsupervised Learning using Nonequilibrium Thermodynamics
Sohl-Dickstein, J., Weiss, E. A., Maheswaranathan, N., & Ganguli, S. (2015) · arXiv · doi:10.48550/arxiv.1503.03585 · Cited by 2,637
Fei-Fei, L., Fergus, R., & Perona, P. (2007) · Computer Vision and Image Understanding · doi:10.1016/j.cviu.2005.09.012 · Cited by 2,359
Probabilistic machine learning and artificial intelligence
Ghahramani, Z. (2015) · Nature · doi:10.1038/nature14541 · Cited by 1,995
Self-citationAleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods
Hüllermeier, E., & Waegeman, W. (2021) · Machine Learning · doi:10.1007/s10994-021-05946-3 · Cited by 1,834
FAB-MAP: Probabilistic Localization and Mapping in the Space of Appearance
Cummins, M., & Newman, P. (2008) · The International Journal of Robotics Research · doi:10.1177/0278364908090961 · Cited by 1,472
A simple new approach to variable selection in regression, with application to genetic fine mapping
Wang, G., Sarkar, A., Carbonetto, P., & Stephens, M. (2020) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/rssb.12388 · Cited by 1,121
Decontamination of ambient RNA in single-cell RNA-seq with DecontX
Yang, S., Corbett, S. E., Koga, Y., Wang, Z., Johnson, W. E., Yajima, M., & Campbell, J. D. (2020) · Genome Biology · doi:10.1186/s13059-020-1950-6 · Cited by 635
Machine learning and AI in marketing–Connecting computing power to human insights
Ma, L., & Sun, B. (2020) · International Journal of Research in Marketing · doi:10.1016/j.ijresmar.2020.04.005 · Cited by 602
Automatic differentiation variational inference
Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., & Blei, D. M. (2016) · arXiv · doi:10.48550/arxiv.1603.00788 · Cited by 515
Stegle, O., Parts, L., Durbin, R., & Winn, J. (2010) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1000770 · Cited by 497
Beal, M. J., & Ghahramani, Z. (2003) · Bayesian Statistics 7 · doi:10.1093/oso/9780198526155.003.0025 · Cited by 458
Self-citationBayesian approach for neural networks—review and case studies
Lampinen, J., & Vehtari, A. (2001) · Neural Networks · doi:10.1016/s0893-6080(00)00098-8 · Cited by 392
Conditional random fields for activity recognition
Vail, D. L., Veloso, M. M., & Lafferty, J. D. (2007) · Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems · doi:10.1145/1329125.1329409 · Cited by 357
Generating Sentences by Editing Prototypes
Guu, K., Hashimoto, T. B., Oren, Y., & Liang, P. (2018) · Transactions of the Association for Computational Linguistics · doi:10.1162/tacl_a_00030 · Cited by 300
A Survey on Bayesian Deep Learning
Wang, H., & Yeung, D.-Y. (2021) · ACM Computing Surveys · doi:10.1145/3409383 · Cited by 247
Active inference on discrete state-spaces: A synthesis
Da Costa, L., Parr, T., Sajid, N., Veselic, S., Neacsu, V., & Friston, K. (2020) · Journal of Mathematical Psychology · doi:10.1016/j.jmp.2020.102447 · Cited by 226
Hybrid physics-informed neural networks for lithium-ion battery modeling and prognosis
Nascimento, R. G., Corbetta, M., Kulkarni, C. S., & Viana, F. A. C. (2021) · Journal of Power Sources · doi:10.1016/j.jpowsour.2021.230526 · Cited by 224
Bayesian Methods for Neural Networks and Related Models
Titterington, D. M. (2004) · Statistical Science · doi:10.1214/088342304000000099 · Cited by 162
Deciphering protein evolution and fitness landscapes with latent space models
Ding, X., Zou, Z., & Brooks III, C. L. (2019) · Nature Communications · doi:10.1038/s41467-019-13633-0 · Cited by 106
Predictive Coding: a Theoretical and Experimental Review
Millidge, B., Seth, A., & Buckley, C. L. (2021) · arXiv · doi:10.48550/arxiv.2107.12979 · Cited by 106
Deep Differentiable Random Forests for Age Estimation
Shen, W., Guo, Y., Wang, Y., Zhao, K., Wang, B., & Yuille, A. (2021) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2019.2937294 · Cited by 91
Wiebe, N., Kapoor, A., & Svore, K. M. (2016) · Quantum Information and Computation · doi:10.26421/qic16.7-8-1 · Cited by 85
Uncertainty-Aware Audiovisual Activity Recognition Using Deep Bayesian Variational Inference
Subedar, M., Krishnan, R., Meyer, P. L., Tickoo, O., & Huang, J. (2019) · 2019 IEEE/CVF International Conference on Computer Vision (ICCV) · doi:10.1109/iccv.2019.00640 · Cited by 85
Diffusion Policies as an Expressive Policy Class for Offline Reinforcement Learning
Wang, Z., Hunt, J. J., & Zhou, M. (2022) · arXiv · doi:10.48550/arxiv.2208.06193 · Cited by 83
“…x 0 ∼ q ( x 0 ) in T steps with a pre-defined variance schedule β i , expressed as A reverse diffusion chain, constructed as p θ ( x 0: T ) := N ( x T ; 0 , I ) , is then optimized by maximizing the evidence lower bound defined as E q [ln p θ ( x 0: T ) q ( x 1: et al., 1999; Blei et al., 2017).”
MethodA theory of learning to infer.
Dasgupta, I., Schulz, E., Tenenbaum, J. B., & Gershman, S. J. (2020) · Psychological Review · doi:10.1037/rev0000178 · Cited by 70
Utilizing Variational Optimization to Learn Markov Random Fields
Tappen, M. F. (2007) · 2007 IEEE Conference on Computer Vision and Pattern Recognition · doi:10.1109/cvpr.2007.383037 · Cited by 54
Subedi, S., & McNicholas, P. D. (2014) · Advances in Data Analysis and Classification · doi:10.1007/s11634-014-0165-7 · Cited by 51
Evaluation Examples are not Equally Informative: How should that change NLP Leaderboards?
Rodriguez, P., Barrow, J., Hoyle, A. M., Lalor, J. P., Jia, R., & Boyd-Graber, J. (2021) · Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers) · doi:10.18653/v1/2021.acl-long.346 · Cited by 47
Meta-learned models of cognition
Binz, M., Dasgupta, I., Jagadish, A. K., Botvinick, M., Wang, J. X., & Schulz, E. (2024) · Behavioral and Brain Sciences · doi:10.1017/s0140525x23003266 · Cited by 38
Dynamic Networks from Hierarchical Bayesian Graph Clustering
Park, Y., Moore, C., & Bader, J. S. (2010) · PLoS ONE · doi:10.1371/journal.pone.0008118 · Cited by 34
Probabilistic inference of Bayesian neural networks with generalized expectation propagation
Zhao, J., Liu, X., He, S., & Sun, S. (2020) · Neurocomputing · doi:10.1016/j.neucom.2020.06.060 · Cited by 20
Learning beyond sensations: How dreams organize neuronal representations
Deperrois, N., Petrovici, M. A., Senn, W., & Jordan, J. (2024) · Neuroscience & Biobehavioral Reviews · doi:10.1016/j.neubiorev.2023.105508 · Cited by 9
Wiebe, N., Kapoor, A., & Svore, K. M. (2014) · arXiv · doi:10.48550/arxiv.1412.3489 · Cited by 5
Metadata
- Authors
- Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K.
- Affiliations
- University of California, Berkeley (US)
- Gatsby Computational Neuroscience Unit (GB)
- Massachusetts Institute of Technology (US)
- AT&T (United States) (US)
- Venue
- Machine Learning
- Published
- 1999-11
- Volume
- 37
- Issue
- 2
- Pages
- 183-233
- ISSN
- 0885-6125 (ISSN-L)
- 1573-0565
- OpenAlex
- W1516111018
- Cited by
- 3,791
- References
- 2
- Citations by year
- 2026: 148 · 2025: 158 · 2024: 132 · 2023: 137 · 2022: 142 · 2021: 244 · 2020: 315 · 2019: 298 · 2018: 217 · 2017: 182 · 2016: 162 · 2015: 182 · 2014: 151 · 2013: 141 · 2012: 173
- FWCI
- 35.87
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Machine Learning (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Physical Sciences › Computer Science › Artificial Intelligence › Gaussian Processes and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Keywords
- variational methods, probabilistic graphical models, Bayesian networks, Markov random fields, approximate inference, convex duality, Boltzmann machine, hidden Markov models, probabilistic learning
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review