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1、5MCMCUsingHamiltonianDynamicsRadfordM.Neal5.1IntroductionMarkovchainMonteCarlo(MCMC)originatedwiththeclassicpaperofMetropolisetal.(1953),whereitwasusedtosimulatethedistributionofstatesforasystemofidealizedmolecules.Notlongafter,anotherapproachtomolecularsimulationwasintroduced(AlderandWainwr
2、ight,1959),inwhichthemotionofthemoleculeswasdeterministic,followingNewtonslawsofmotion,whichhaveanelegantformalizationasHamiltoniandynamics.Forfindingthepropertiesofbulkmaterials,theseapproachesareasymptoticallyequivalent,sinceeveninadeterministicsimulation,eachlocalregionofthematerialexperie
3、nceseffectivelyrandominfluencesfromdistantregions.Despitethelargeoverlapintheirappli-cationareas,theMCMCandmoleculardynamicsapproacheshavecontinuedtocoexistinthefollowingdecades(seeFrenkelandSmit,1996).In1987,alandmarkpaperbyDuane,Kennedy,Pendleton,andRowethunitedtheMCMCandmoleculardynamicsap
4、proaches.TheycalledtheirmethodhybridMonteCarlo,whichabbreviatestoHMC,butthephraseHamiltonianMonteCarlo,retain-ingtheabbreviation,ismorespecificanddescriptive,andIwilluseithere.Duaneetal.appliedHMCnottomolecularsimulation,buttolatticefieldtheorysimulationsofquan-tumchromodynamics.Statisticalapp
5、licationsofHMCbeganwithmyuseofitforneuralnetworkmodels(Neal,1996a).Ialsoprovidedastatistically-orientedtutorialonHMCinareviewofMCMCmethods(Neal,1993,Chapter5).TherehavebeenotherapplicationsofHMCtostatisticalproblems(e.g.Ishwaran,1999;Schmidt,2009)andstatistically-orientedreviews(e.g.Liu,2001
6、,Chapter9),butHMCstillseemstobeunderappreciatedbystatisticians,andperhapsalsobyphysicistsoutsidethelatticefieldtheorycommunity.ThisreviewbeginsbydescribingHamiltoniandynamics.Despiteterminologythatmaybeunfamiliaroutsidephysics,thefeaturesofHamiltoniandynamicsthatareneededforHMCareelementary.T
7、hedifferentialequationsofHamiltoniandynamicsmustbedis-cretizedforcomputerimplementation.Theleapfrogschemethatistypicallyusedisquitesimple.FollowingthisintroductiontoHamiltoniandynamics,Idescribehowtouseittocon-structanMCMCmethod.Thefirststepistodefin
