localsystemx

UsingMCMCSeparatingMCMCfromBayesianInferenceLinefittingrevisitedAtoyequaliserproblemSomelessonsAprobleminfilmrestoration/retouchingMCMCis(just)atoolChooseaModelIdentifyParametersArticulateProbabilities[BayesianInference]TrytoseeifyoucanintegrateoutnuisancesDerivethePosteriorSolveDeterministicallyDirect,CG,SteepestDescentetcUseMCMCManipulateRandomSamplesTogetoneanswer(ifyouwant)GivesyouoneanswerNeedbettermodelIfsolutionnotokGood,BadCanalwaysdesignsingleparameter-at-a-timeschemes.
SoiterationscanbeverylowcomplexitySimpleiterations=longconvergenceGivesyouapictureofalternateanswersDoyoureallyneedalternateanswersWillalwaysallowyoutogetto"best"solutionIterationscanbehighcomplexityConvergencecanberapid(e.
g.
CG)forwelldefinedproblemsGivesyoujustoneanswerCangivelocalminimumfornon-linearproblemsMCMCOthersUglyTosolveyourproblemyouneedagoodmodelMCMCisnotreallygoingtohelpyouifyouhavethewrongmodelMCMCsuitedtoBIGproblems:butwhatisBIGreallyE.
g.
Exhaustivesearchformotionestimationispossibleinrealtime(TVrates)inhardware:whybotherwithotherthings(anexaggeration…butinterestingnevertheless)LineFitting(again)NeedsLatexObservedDataActualLineInitialGuessTypicalResultsNiceConvergencebecausewecandrawsamplesdirectlySeeMatlabdemoTypicalResultscmvar_eWatchoutAllrandomnumbergeneratorsarenotcreatedequal(SeeNR)Harderproblemsrequirelongerruns(ofcourse)SometimeshardtogetallbugsoutbecauseitsallarandomsearchanywayBlind()EqualisationSignal2ndOrderAllpoleSystemNoiseRec'dSignalIdentifythesystemcoefficientsANDrecovertheoriginalsignalComms,Deblurring,OvershootCancellationEqualiserProblemNowmorelatexDirectnumericalsampling010.
30.
250.
20.
251234NumberlineinterpretationP(1)=0.
3,p(2)=0.
25,P(3)=0.
2,p(4)=0.
250.
7571pointsevaluatedGibbssampler1(equaliser1.
m)TypicalEstimatedSystemActualSystemX20!
300iterationsBacktoLatexGibbsSamplerII(equaliser2.
m)UsingFilterBank(systemchoices)30filtersSamplesfromfilterbankSamplesofsignalparameterSystemEstimateEqualisedsignalLessonsGibbssamplertakesbigproblemsandbreaksthemintolotsofsmallonesSpottingthefunctionalformofaknownp.
d.
f.
isausefulskill.
Bookshelp.
Ifallelsefails,canalwayssampledirectlyMCMCdoesnotnecessarilysolveyourproblem.
Goodpriors,bettermodelsarestillimportantDeterministic/StochasticHybridmixisv.
useful