Advanced Topics in MCMC

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1、27AdvancedTopicsinMCMC27.1IntroductionThepaceofresearchonMCMCmethodsissoquickthatanysurveyofadvancedtopicsisimmediatelyobsolete.ThehighlyeclecticanddecidedlybiasedcoverageinourfinalchapterbeginswithadiscussionofMarkovrandomfields.Ourlimitedaimsherearetoprovethe

2、Hammersley-CliffordtheoremandintroducetheSwendsen-Wangalgorithm,acleverformofslicesampling.IntheIsingmodel,theSwendsen-WangalgorithmismuchmoreefficientthanstandardGibbssampling.WethenmoveontoreversiblejumpMCMC.Bayesianinferencetendstoperformbetterthanfrequentist

3、inferenceinchoosingbetweennon-nestedmodels.Green’s[8]inventionofreversiblejumpMCMCprovidesapracticalpathtoimplementingBayesianmodelselection.Inouropinion,reversiblejumpMCMCwillhavealargerimpactonthefieldofstatisticsthanthemoreglamorousdeviceofperfectsampling.T

4、hesecondhalfofthechapterisdesignedtohelpreaderssortthroughtheadvancedliteratureonconvergenceratesofMarkovchains.Ourtreat-mentofconvergencecommenceswiththeintroductionofthetotalvariationandchi-squaredistances.Couplingismentionedbecauseitoftenyieldssur-prisingl

5、ygoodtotalvariationboundswithlittleeffort.Convergencetheoryismucheasierforfinite-stateMarkovchainsthanitisfordenumerableandcontinuouschains.Forreversiblefinite-statechains,thefullforceoflinearalgebracanbebroughttobear.Thisisagoodsettingtodevelopintuition.Forthes

6、akeofcomparison,wetreatconvergenceoftheindependencesam-plerfromthedifferentperspectivesoflinearalgebraandcoupling.ThelasttwosectionsofthechapteremphasizeHilbertspaces,compactoperators,ouroldfriendthesingularvaluedecomposition,andorthogonalpolynomials.Thefounda

7、tionsofHilbertandBanachspaceswerelaidinthenineteenthcenturyandcompletedinthefirsthalfofthetwentiethcentury.Hilbertspacesisolateandgeneralizetheinnerproductsthatdrivelinearalgebra.Linearoperatorssuchasexpectationsandconditionalexpectationsaretheinfinite-dimensio

8、nalanalogsofmatrices.Throughoutourdiscussion,alertreaderswillnoticeourdebttothebooks[11,13,15]andthesurveyarticle[6].Thelovelyinterplayofdifferentbranchesofmathematicsyieldsenormousinsight