AI帮你理解科学

AI 生成解读视频

AI抽取解析论文重点内容自动生成视频

生成解读视频

AI 溯源

AI解析本论文相关学术脉络

生成溯源树

AI 精读

AI抽取本论文的概要总结

微博一下：

We have described a variety of applications of variational methods to problems of inference and learning in graphical models

An introduction to variational methods for graphical models

Learning in graphical models, no. 2 (1999): 183-233

引用：3850|浏览113

下载 PDF 全文

原文链接

其它链接

academic.microsoft.com link.springer.com dl.acm.org dblp.uni-trier.de

引用

微博一下

关键词

graphical models Bayesian networks belief networks probabilistic inference approximate inference更多(5+)

摘要

This paper presents a tutorial introduction to the use of variational methods for inference and learning in graphical models (Bayesian networks and Markov random fields). We present a number of examples of graphical models, including the QMR-DT database, the sigmoid belief network, the Boltzmann machine, and several variants of hidden Mar...更多

代码：

数据：

精读

下载 PDF 全文

PPT (上传PPT)

相似论文

引用论文

被引论文

简介

The problem of probabilistic inference in graphical models is the problem of computing a conditional probability distribution over the values of some of the nodes, given the values of other nodes.
The authors often wish to calculate marginal probabilities in graphical models, in particular the probability of the observed evidence, P(E).
Inference algorithms do not compute the numerator and denominator of Eq (1) and divide, they generally produce the likelihood as a by-product of the calculation of P(HIE).
Algorithms that maximize likelihood generally make use of the calculation of P(HIE) as a subroutine

重点内容

The problem of probabilistic inference in graphical models is the problem of computing a conditional probability distribution over the values of some of the nodes, given the values of other nodes
Viewed as a function of the parameters of the graphical model, for fixed E, P(E) is an important quantity known as the likelihood
We provide a brief overview of the QMR-DT database here; for further details see Shwe, et al (1991)
As in the case of the Boltzmann machine, we find that the variational parameters are linked via their Markov blankets and the consistency equation (Eq (67)) can be interpreted as a local message-passing algorithm
We have described a variety of applications of variational methods to problems of inference and learning in graphical models
It is important to emphasize, that research on variational methods for graphical models is of quite recent origin, and there are many open problems and unresolved issues

方法

The authors make a few remarks on the relationships between variational methods and stochastic methods, in particular the Gibbs sampler.
In Gibbs sampling, the message-passing is simple: each node learns the current instantiation of its Markov blanket.
With enough samples the node can estimate the distribution over its Markov blanket and determine its own statistics.
The authors can quite generally treat parameters as additional nodes in a graphical model (cf.
This volume) and thereby treat Bayesian inference on the same footing as generic probabilistic inference in a graphical model
This probabilistic inference problem is often intractable, and variational approximations can be useful.
The ensemble is fit by minimizing the appropriate KL divergence:

结论

The authors have described a variety of applications of variational methods to problems of inference and learning in graphical models.
The authors hope to have convinced the reader that variational methods can provide a powerful and elegant tool for graphical models, and that the algorithms that result are simple and intuitively appealing.
It is important to emphasize, that research on variational methods for graphical models is of quite recent origin, and there are many open problems and unresolved issues.

引用论文

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.
Cover, T., & Thomas, J. (1991). Elements of Information Theory. New York: John Wiley.
Cowell, R. (in press). Introduction to inference for Bayesian networks. In M. I. Jordan (Ed.), Learning in Graphical Models. Norwell, MA: Kluwer Academic Publishers.
Dagum, P., & Luby, M. (1993). Approximating probabilistic inference in Bayesian belief networks is NP-hard. Artificial Intelligence, 60, 141-153.
Dayan, P., Hinton, G. E., Neal, R., & Zemel, R. S. (1995). The Helmholtz Machine.
Dean, T., & Kanazawa, K. (1989). A model for reasoning about causality and persistence.
Dechter, R. (in press). Bucket elimination: A unifying framework for probabilistic inference. In M. I. Jordan (Ed.), Learning in Graphical Models. Norwell, MA: Kluwer
Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum-likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B39, 1-38.
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.
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..
Ghahramani, Z., & Hinton, G. E. (1996). Switching state-space models. University of
Ghahramani, Z., & Jordan, M. I. (1997). Factorial Hidden Markov models. Machine
Gilks, W., Thomas, A., & Spiegelhaiter, D. (1994). A language and a program for complex Bayesian modelling. The Statistician, 43, 169-178.
Heckerman, D. (in press). A tutorial on learning with Bayesian networks. In M. I. Jordan (Ed.), Learning in Graphical Models. Norwell, MA: Kluwer Academic Publishers.
Henrion, M. (1991). Search-based methods to bound diagnostic probabilities in very large belief nets. Uncertainty and Artificial Intelligence: Proceedings of the Seventh
Hinton, G. E., & Sejnowski, T. (1986). Learning and relearning in Boltzmann machines. In
D. E. Rumelhart & J. L. McClelland, (Eds.), Parallel distributed processing: Volume
Hinton, G.E. & van Camp, D. (1993). Keeping neural networks simple by minimizing the description length of the weights. In Proceedings of the 6th Annual Workshop on
Hinton, G. E., Dayan, P., Frey, B., and Neal, R. M. (1995). The wake-sleep algorithm for unsupervised neural networks. Science, 268:1158-116l.
Hinton, G. E., Sallans, B., & Ghahramani, Z. (in press). A hierarchical community of experts. In M. I. Jordan (Ed.), Learning in Graphical Models. Norwell, MA: Kluwer
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.1. (1996). Computing upper and lower bounds on likelihoods in intractable networks. Uncertainty and Artificial Intelligence: Proceedings of the
Jaakkola, T. S. (1997). Variational methods for inference and estimation in graphical models. Unpublished doctoral dissertation, Massachusetts Institute of Technology.
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 g. 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
Jaakkola, T. S., & Jordan. M.1. (1997c). Variational methods and the QMR-DT database.
Submitted to: Journal of Artificial Intelligence Research.
Jaakkola, T. S., & Jordan. M. I. (in press). Improving the mean field approximation via the use of mixture distributions. In M. I. Jordan (Ed.), Learning in Graphical Models.
Jensen, C. S., Kong, A., & Kjrerulff, U. (1995). Blocking-Gibbs sampling in very large probabilistic expert systems. International Journal of Human-Computer Studies, 42, 647-666.
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
Kanazawa, K., Koller, D., & RusselJ, S. (1995). Stochastic simulation algorithms for dynamic probabilistic networks. Uncertainty and Artificial Intelligence: Proceedings of the Eleventh Conference. San Mateo, CA: Morgan Kaufmann.
Kjrerulff, U. (1990). Triangulation of graphs-algorithms giving small total state space.
Kjrerulff, 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. (1997a). Ensemble learning for hidden Markov models. Unpublished manuscript. Department of Physics, University of Cambridge.
MacKay, D.J.C. (1997b). Comparison of approximate methods for handling hyperparameters. Submitted to Neural Computation.
MacKay, D.J.C. (1997b). Introduction to Monte Carlo methods. In M. I. Jordan (Ed.), Learning in Graphical Models. Norwell, MA: Kluwer Academic Publishers.
McEliece, R.J., MacKay, D.J.C., & Cheng, J.-F. (1996) Turbo decoding as an instance of Pearl's "belief propagation algorithm." Submitted to: IEEE Journal on Selected Areas in Communication.
Merz, C. J., & Murphy, P. M. (1996). UCI repository of machine learning databases. [http:/wwv.ics.uci/......mlearn/MLRepository.html]. Irvine, CA: University of California, Department of Information and Computer Science.
Neal, R. (1992). Connectionist learning of belief networks, Artificial Intelligence, 56, 71113.
Neal, R. (1993). Probabilistic inference using Markov chain Monte Carlo methods. University of Toronto Technical Report CRG-TR-93-1, Department of Computer Science.
Neal, R., & Hinton, G. E. (in press). A view of the EM algorithm that justifies incremental, sparse, and other variants. In M. I. Jordan (Ed.), Learning in Graphical Models. Norwell, MA: Kluwer Academic Publishers.
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). Modem 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-'~6.
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.
Saul, L. K., & Jordan, M. I. (in press). A mean field learning algorithm for unsupervised neural networks. In M. I. Jordan (Ed.), Learning in Graphical Models. Norwell, MA: Kluwer Academic Publishers.
Seung, S. (1995). Annealed theories oflearning. In J.-H Oh, C. Kwon, and 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.
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-l/QMR lmowledge base. Meth. Inform. Med., 90, 241-255.
Smyth, P., Heckerman, D., & Jordan, M. I. (1997). Probabilistic independence networks for hidden Markov probability models. Neural Computation, 9, 227-270.
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.
Williams, C. K. I., & Hinton, G. E. (1991). Mean field networks that learn to discriminate temporally distorted strings. In Touretzky, D. S., Elman, J., Sejnowski, T., & Hinton, G. E., (Eds.), Proceedings of the 1990 Connectionist Models Summer School. San Mateo, CA: Morgan Kaufmann.