graphical models and variational methods

[Michael I. Jordan]

A tutorial paper on graphical models and variational approximations 
is available at:

    ftp://psyche.mit.edu/pub/jordan/variational-intro.ps.gz
    http://www.ai.mit.edu/projects/jordan.html


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	  AN INTRODUCTION TO VARIATIONAL METHODS
       		  FOR GRAPHICAL MODELS

	 	    Michael I. Jordan
	  Massachusetts Institute of Technology

		    Zoubin Ghahramani
	          University of Toronto

		    Tommi S. Jaakkola
	   University of California Santa Cruz

		     Lawrence K. Saul
	          AT&T Labs -- Research


  This paper presents a tutorial introduction to the use of 
  variational methods for inference and learning in graphical 
  models.  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 Markov 
  models, in which it is infeasible to run exact inference 
  algorithms.  We then introduce variational methods, showing 
  how upper and lower bounds can be found for local probabilities, 
  and discussing methods for extending these bounds to bounds 
  on global probabilities of interest.  Finally we return to 
  the examples and demonstrate how variational algorithms can 
  be formulated in each case.