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1、MCMCmethodsforBayesianInferenceA.TaylanCemgilSignalProcessingandCommunicationsLab.5R1StochasticProcessesMarch06,2008Cemgil5R1StochasticProcesses,MCMCmethodsforBayesianInference.March06,2008OutlineGoal:ProvidemotivatingexamplestothetheoryofMarkovchains(thatSum
2、eetSinghhascovered)?BayesianInference,ProbabilitymodelsandGraphicalmodelnotation?TheGibbssampler?Metropolis-Hastings,MCMCTransitionKernels,?Sketchofconvergenceresults?SimulatedannealinganditerativeimprovementCemgil5R1StochasticProcesses,MCMCmethodsforBayesian
3、Inference.March06,20081Bayes’TheoremThomasBayes(1702-1761)“WhatyouknowaboutaparameterλafterthedataDarriveiswhatyouknewbeforeaboutλandwhatthedataDtoldyou1.”p(D
4、λ)p(λ)p(λ
5、D)=p(D)Likelihood×PriorPosterior=Evidence1(Janes2003(ed.byBretthorst);MacKay2003)Cemgil5R1
6、StochasticProcesses,MCMCmethodsforBayesianInference.March06,20082AnapplicationofBayes’Theorem:“SourceSeparation”Giventwofairdicewithoutcomesλandy,D=λ+yWhatisλwhenD=9?Cemgil5R1StochasticProcesses,MCMCmethodsforBayesianInference.March06,20083“Burocratical”deriv
7、ationFormallywewritep(λ)=C(λ;[1/61/61/61/61/61/6])p(y)=C(y;[1/61/61/61/61/61/6])p(D
8、λ,y)=δ(D?(λ+y))1x=0Kroneckerdeltafunctiondenotingadegenerate(deterministic)distributionδ(x)=0x6=01p(λ,y
9、D)=×p(D
10、λ,y)×p(y)p(λ)p(D)1Posterior=×Likelihood×PriorEvidenceXp(λ
11、D)=p
12、(λ,y
13、D)PosteriorMarginalyCemgil5R1StochasticProcesses,MCMCmethodsforBayesianInference.March06,20084AnapplicationofBayes’Theorem:“SourceSeparation”D=λ+y=9D=λ+yy=1y=2y=3y=4y=5y=6λ=1234567λ=2345678λ=3456789λ=45678910λ=567891011λ=6789101112Bayestheorem“upgrades”p
14、(λ)intop(λ
15、D).Butyouhavetoprovideanobservationmodel:p(D
16、λ)Cemgil5R1StochasticProcesses,MCMCmethodsforBayesianInference.March06,20085AnotherapplicationofBayes’Theorem:“ModelSelection”Givenanunknownnumberoffairdicewithoutcomesλ1,λ2,...,λn,XnD=λii=1Howmanydicear
17、etherewhenD=9?AssumethatanynumbernisequallylikelyCemgil5R1StochasticProcesses,MCMCmethodsforBayesianInference.March06,20086AnotherapplicationofBayes’Theorem:“ModelSelection”Givenallnareeq
