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1、AlexanderV.Mantzaris(2008/2009)BasicIntroductioninto-MCMC(MarkovChainMonteCarlo)*ConsideredtobeoneofthetoptenmostimportantalgorithmseverToseehowgreatMCMCis,wewilllookatthemotivationforsamplingfirst,andthenthemethodsusedbeforeitsintroductionMotivationofsampling•Manyfunctio
2、ns,Evensuchasimplefunctioncannotbeequations,andintegrated,withoutnumericalmethods.distributionscannotInthefieldofprobability,integrals/summationsbeintegrated(continuous/discreterespectively),arevitalforcalculatingtheexpectationoranalytically.Forexpectedvaluesofdistributio
3、ns.example:Andthisisessentiallywhatmarginalisationisallaboutaswell.2xMotivationofsampling•Drawingstatisticallyconsistentsamplesfromadistribution-Bydrawingsamplesconsistenttoyourdistribution,youareeffectivelycreatingasimulation.Whichcanbeusefulforanalyzinghowfastthemeans,v
4、ariances,etc,progress,andifthedistribution’ssamplesarerequiredasinputintoanotherdistribution.•WesamplewhentheCumulativeDistributionFunction(CDF)cannotbefoundanalytically.SamplingfromtheCDFishowMatlabcomesupwithrandomsamplesfordistributionsliketheGaussian…NextslideMotivati
5、onofsamplingWikipedia•Withnormaliseddistributionstheintegral=1,andsamplingrandomnumbersin[0,1]tothenmapthemfromtheCDFy-axistotheCDFx-axis,givesusthepointwhereasampleischosenOntheoriginaldistribution,wheretheheightislargertheslopeontheCDFishigherexposingittogreaterchanceso
6、farandomnumbergettingchosenonthey-axisMonteCarlo•ThemostwidelyusedplaceofMonteCarlosamplingisinMonteCarloNrandomsamplesalongIntegrationthedomain•Usesrandomnesstocomeupwitharandomvariableestimates,similartox,x,...,x12Nthegamblingprocessincasinos,wherethenamederives.•Define
7、ddomainalongwhichweTheaverageofthesample,uniformly(non-uniformisaindependentsamplespriorhardertoincorporatehere),andN1sinceeachpointwastakenwithE(f;N)=∑f(x)equalprobability,theintegraliNi=1(expectationforrandomvariables),issimplytheaverageofthesamples.•Eachsampleisindepen
8、dentlysampledMonteCarloFromthelawoflargenumbers,asNincreasestheconfidenceintheestimateprovideinc
