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TR2001-16 May 2001 · This is an updated and expanded version of TR2000-26,but is still in draft...

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MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Bethe free energy, Kikuchi approximations, and belief propagation algorithms Jonathan S. Yedidia, William T. Freeman, Yair Weiss TR2001-16 May 2001 Abstract This is an updated and expanded version of TR2000-26, but it is still in draft form. More im- portantly, our analysis lets us build on the progress made in statistical physics since Bethe’s approximation was introduced in 1935. Kikuchi and others have shown how to construct more accurate free energy approximations, of which Bethe’s approximation is the simplest. Exploit- ing the insights from our analysis, we derive generalized belief propagation (GBP) versions of these Kikuchi approximations. These new message passing algorithms can be significantly more accurate than ordinary BP, at an adjustable increase in complexity. We illustrate such a new GBP algorithm on a grid Markov network and show that it gives much more accurate marginal probabilities than those found using ordinary BP. This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories, Inc., 2001 201 Broadway, Cambridge, Massachusetts 02139
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Page 1: TR2001-16 May 2001 · This is an updated and expanded version of TR2000-26,but is still in draft form as of its release on April 4, 2001. Small further update (section 2.3) on May

MITSUBISHI ELECTRIC RESEARCH LABORATORIEShttp://www.merl.com

Bethe free energy, Kikuchi approximations,and belief propagation algorithms

Jonathan S. Yedidia, William T. Freeman, Yair Weiss

TR2001-16 May 2001

Abstract

This is an updated and expanded version of TR2000-26, but it is still in draft form. More im-portantly, our analysis lets us build on the progress made in statistical physics since Bethe’sapproximation was introduced in 1935. Kikuchi and others have shown how to construct moreaccurate free energy approximations, of which Bethe’s approximation is the simplest. Exploit-ing the insights from our analysis, we derive generalized belief propagation (GBP) versions ofthese Kikuchi approximations. These new message passing algorithms can be significantly moreaccurate than ordinary BP, at an adjustable increase in complexity. We illustrate such a newGBP algorithm on a grid Markov network and show that it gives much more accurate marginalprobabilities than those found using ordinary BP.

This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in partwithout payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies includethe following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment ofthe authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, orrepublishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. Allrights reserved.

Copyright c©Mitsubishi Electric Research Laboratories, Inc., 2001201 Broadway, Cambridge, Massachusetts 02139

Page 2: TR2001-16 May 2001 · This is an updated and expanded version of TR2000-26,but is still in draft form as of its release on April 4, 2001. Small further update (section 2.3) on May

MERLCoverPageSide2

Page 3: TR2001-16 May 2001 · This is an updated and expanded version of TR2000-26,but is still in draft form as of its release on April 4, 2001. Small further update (section 2.3) on May

MITSUBISHI ELECTRICRESEARCHLABORATORIEShttp://www.merl.com

Bethe free energy, Kikuchi approximations,and belief propagation algorithms

JonathanS.Yedidia,William T. Freeman,andYair Weiss

TR-2001-16 May 2001

Abstract

This is anupdatedandexpandedversionof TR2000-26,but it is still in draft form.Beliefpropagation(BP)wasonlysupposedtowork for tree-likenetworksbutworks

surprisinglywell in many applicationsinvolving networkswith loops,includingturbocodes.However, therehasbeenlittle understandingof thealgorithmor thenatureofthesolutionsit findsfor generalgraphs.

We show that BP canonly converge to a stationarypoint of an approximatefreeenergy, known astheBethefreeenergy in statisticalphysics.This resultcharacterizesBP fixed-pointsand makes connectionswith variationalapproachesto approximateinference.

More importantly, our analysislets us build on the progressmadein statisticalphysicssinceBethe’s approximationwasintroducedin 1935.Kikuchi andothershaveshown how to constructmoreaccuratefree energy approximations,of which Bethe’sapproximationis the simplest. Exploiting the insightsfrom our analysis,we derivegeneralizedbeliefpropagation(GBP)versionsof theseKikuchi approximations.Thesenew messagepassingalgorithmscanbesignificantlymoreaccuratethanordinaryBP,at anadjustableincreasein complexity. We illustratesucha new GBPalgorithmon agrid Markov network andshow thatit givesmuchmoreaccuratemarginalprobabilitiesthanthosefoundusingordinaryBP.

This work may not be copiedor reproducedin whole or in part for any commercialpurpose.Permissionto copy inwholeor in partwithoutpaymentof feeis grantedfor nonprofiteducationalandresearchpurposesprovidedthatall suchwholeor partialcopiesincludethefollowing: anoticethatsuchcopying is by permissionof MitsubishiElectricResearchLaboratories,Inc.;anacknowledgmentof theauthorsandindividualcontributionsto thework; andall applicableportionsof thecopyright notice.Copying, reproduction,or republishingfor any otherpurposeshallrequirealicensewith paymentof feeto MitsubishiElectricResearchLaboratories,Inc. All rightsreserved.

Page 4: TR2001-16 May 2001 · This is an updated and expanded version of TR2000-26,but is still in draft form as of its release on April 4, 2001. Small further update (section 2.3) on May

Copyright c�

MitsubishiElectricResearchLaboratories,Inc., 2001201Broadway, Cambridge,Massachusetts02139

Page 5: TR2001-16 May 2001 · This is an updated and expanded version of TR2000-26,but is still in draft form as of its release on April 4, 2001. Small further update (section 2.3) on May

This is anupdatedandexpandedversionof TR2000-26,but is still in draft form asof its releaseonApril 4, 2001.Smallfurtherupdate(section2.3)onMay 15,2001.

Page 6: TR2001-16 May 2001 · This is an updated and expanded version of TR2000-26,but is still in draft form as of its release on April 4, 2001. Small further update (section 2.3) on May

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