Bayesian machine learning - DataRobot AI Cloud

Bayesian ML is a paradigm for constructing statistical models based on Bayes' Theorem. Learn more from the experts at DataRobot. Skiptocontent Blog Bayesianmachinelearning Bayesianmachinelearning September3,2020 by DataRobot · 7min read ThisarticlewasoriginallypublishedatAlgorithimia’swebsite.ThecompanywasacquiredbyDataRobotin2021.Thisarticlemaynotbeentirelyup-to-dateorrefertoproductsandofferingsnolongerinexistence.FindoutmoreaboutDataRobotMLOpshere. BayesianMLisaparadigmforconstructingstatisticalmodelsbasedonBayes’Theorem.LearnmorefromtheexpertsatDataRobot. Thinkaboutastandardmachinelearningproblem.Youhaveasetoftrainingdata,inputsandoutputs,andyouwanttodeterminesomemappingbetweenthem.So,yousplicetogetheramodelandsoonyouhaveadeterministicwayofgeneratingpredictionsforatargetvariable 𝑦y givenanunseeninput 𝑥x.  There’sjustoneproblem–youdon’thaveanywaytoexplainwhat’sgoingonwithinyourmodel!Allyouknowisthatit’sbeentrainedtominimizesomelossfunctiononyourtrainingdata,butthat’snotmuchtogoon.Ideally,you’dliketohaveanobjectivesummaryofyourmodel’sparameters,completewithconfidenceintervalsandotherstatisticalnuggets,andyou’dliketobeabletoreasonaboutthemusingthelanguageofprobability.  That’swhereBayesianMachineLearningcomesin. WhatisBayesianmachinelearning? BayesianMLisaparadigmforconstructingstatisticalmodelsbasedonBayes’Theorem p(θ|x)=p(x|θ)p(θ)p(x) Generallyspeaking,thegoalofBayesianMListoestimatetheposteriordistribution(𝑝(𝜃|𝑥)p(θ|x))giventhelikelihood(𝑝(𝑥|𝜃)p(x|θ))andthepriordistribution, 𝑝(𝜃)p(θ).Thelikelihoodissomethingthatcanbeestimatedfromthetrainingdata.  Infact,that’sexactlywhatwe’redoingwhentrainingaregularmachinelearningmodel.We’reperformingMaximumLikelihoodEstimation,aniterativeprocesswhichupdatesthemodel’sparametersinanattempttomaximizetheprobabilityofseeingthetrainingdata 𝑥x havingalreadyseenthemodelparameters 𝜃θ. SohowdoestheBayesianparadigmdiffer?Well,thingsgetturnedontheirheadinthatinthisinstanceweactuallyseektomaximizetheposteriordistributionwhichtakesthetrainingdataasfixedanddeterminestheprobabilityofanyparametersetting 𝜃θ giventhatdata.Wecallthisprocess MaximumaPosteriori(MAP).It’seasier,however,tothinkaboutitintermsofthelikelihoodfunction.ByBayes’Theoremwecanwritetheposterioras p(θ|x)∝p(x|θ)p(θ) Hereweleaveoutthedenominator, 𝑝(𝑥)p(x),becausewearetakingthemaximizationwithrespectto 𝜃θ which 𝑝(𝑥)p(x) doesnotdependon.Therefore,wecanignoreitinthemaximizationprocedure.ThekeypieceofthepuzzlewhichleadsBayesianmodelstodifferfromtheirclassicalcounterpartstrainedbyMLEistheinclusionoftheterm 𝑝(𝜃)p(θ).Wecallthisthe priordistribution over 𝜃θ.  Theideaisthatitspurposeistoencodeourbeliefsaboutthemodel’sparametersbeforewe’veevenseenthem.That’stosay,wecanoftenmakereasonableassumptionsaboutthe“suitability”ofdifferentparameterconfigurationsbasedsimplyonwhatweknowabouttheproblemdomainandthelawsofstatistics.Forexample,it’sprettycommontouseaGaussianprioroverthemodel’sparameters.Thismeansweassumethatthey’redrawnfromanormaldistributionhavingsomemeanandvariance.Thisdistribution’sclassicbell-curvedshapeconsolidatesmostofitsmassclosetothemeanwhilevaluestowardsitstailsareratherrare.  Byusingsuchaprior,we’reeffectivelystatingabeliefthatmostofthemodel’sweightswillfallinsomenarrowrangeaboutameanvaluewiththeexceptionofafewoutliers,andthisisprettyreasonablegivenwhatweknowaboutmostrealworldphenomena. ItturnsoutthatusingthesepriordistributionsandperformingMAPisequivalenttoperformingMLEintheclassicalsense alongwiththeadditionofregularization.There’saprettyeasymathematicalproofofthisfactthatwewon’tgointohere,butthegististhatbyconstrainingtheacceptablemodelweightsviathepriorwe’reeffectivelyimposingaregularizer. MLOps101:TheFoundationforYourAIStrategy MethodsofBayesianML MAP WhileMAPisthefirststeptowardsfullyBayesianmachinelearning,it’sstillonlycomputingwhatstatisticianscalla pointestimate,thatistheestimateforthevalueofaparameteratasinglepoint,calculatedfromdata.Thedownsideofpointestimatesisthattheydon’ttellyoumuchaboutaparameterotherthanitsoptimalsetting.Inreality,weoftenwanttoknowotherinformation,likehowcertainwearethataparameter’svalueshouldfallwithinthispredefinedrange.  Tothatend,thetruepowerofBayesianMLliesinthecomputationoftheentireposteriordistribution.Thisisatrickybusinessthough.Distributionsarenotnicelypackagedmathematicalobjectsthatcanbemanipulatedatwill.Oftentheycomedefinedtousastricky,intractableintegralsovercontinuousparameterspacesthatareinfeasibletoanalyticallycompute.Therefore,anumberoffascinatingBayesianmethodshavebeendevisedthatcanbeusedto sample (i.e.drawsamplevalues)fromtheposteriordistribution.  MCMC Probablythemostfamousoftheseisanalgorithmcalled MarkovChainMonteCarlo,anumbrellawhichcontainsanumberofsubsidiarymethodssuchas Gibbs and SliceSampling. ThemathbehindMCMCisdifficultbutintriguing.Inessence,thesemethodsworkbyconstructingaknownMarkovchainwhichsettlesintoadistributionthat’sequivalenttotheposterior.AnumberofsuccessoralgorithmsimproveontheMCMCmethodologybyusinggradientinformationtoallowthesamplertomoreefficientlynavigatetheparameterspace. MCMCanditsrelativesareoftenusedasacomputationalcoginabroaderBayesianmodel.Theirdownsideisthattheyareoftenverycomputationallyinefficient,althoughthisdrawbackhasbeenimprovedtremendouslyinrecentyears.Thatsaid,it’softenpreferabletousethesimplesttoolpossibleforanygivenjob.  Tothatend,thereexistmanysimplermethodswhichcanoftengetthejobdone.Forexample,thereexistBayesianlinearandlogisticregressionequivalentsinwhichsomethingcalledthe LaplaceApproximation isused.Thisalgorithmprovidesananalyticalapproximationtotheposteriordistributionbycomputingasecond-orderTaylorexpansionaroundthelog-posteriorandcenteredattheMAPestimate. Gaussianprocess OnepopularBayesianmethodcapableofperformingbothclassificationandregressionisthe Gaussianprocess.AGPisastochasticprocesswithstrictGaussianconditionsimposeduponitsconstituentrandomvariables.GPshavearatherprofoundtheoreticalunderpinning,andmuchefforthasbeendevotedtotheirstudy.Ineffect,theseprocessesprovidetheabilitytoperformregressioninfunctionspace.  Thatis,insteadofchoosingasinglelinetobestfityourdata,youcandetermineaprobabilitydistributionoverthe spaceofallpossiblelines andthenselectthelinethatismostlikelygiventhedataastheactualpredictor.ThisisBayesianestimationinthetruestsenseinthatthefullposteriordistributionisanalyticallycomputed.Theabilitytoactuallyworkoutthemethodinthisinstanceisduetothesuitabilityof conjugatefunctions.  InthecaseofclassificationusingGPs,theposteriorisonceagainnolongerconjugatetothelikelihood,andtheabilitytodoanalyticcomputationsbreaksdown.Inthiscase,it’snecessarytoonceagainresorttoapproximatesolversliketheLaplaceApproximationinordertosuitablytrainthemodeltoadesiredlevelofaccuracy. ThepresenceofBayesianmodelsinML WhileBayesianmodelsarenotnearlyaswidespreadinindustryastheircounterparts,theyarebeginningtoexperienceanewresurgenceduetotherecentdevelopmentofcomputationallytractablesamplingalgorithms,greateraccesstoCPU/GPUprocessingpower,andtheirdisseminationinarenasoutsideacademia.  Theyareespeciallyusefulinlow-datadomainswheredeeplearningmethodsoftenfailandinregimesinwhichtheabilitytoreasonaboutamodelisessential.Inparticular,theyseewideuseinBioinformaticsandHealthcare,asinthesefieldstakingapointestimateatfacevaluecanoftenhavecatastrophiceffectsandmoreinsightintotheunderlyingmodelisnecessary.  Forexample,onewouldnotwanttonaivelytrusttheoutputsofanMRIcancerpredictionmodelwithoutatleastfirsthavingsomeknowledgeabouthowthatmodelwasoperating.Similarly,withinbioinformatics,variantcallerssuchasMutect2andStrelkarelyheavilyonBayesianmethodsunderneaththehood.  ThesesoftwareprogramstakeinDNAreadsfromaperson’sgenomeandlabel“variant”alleleswhichdifferfromthoseinthereference.Inthisdomain,accuracyandstatisticalsoundnessareparamount,soBayesianmethodsmakealotofsensedespitethecomplexitythattheyaddinimplementation.  Overall,BayesianMLisafastgrowingsubfieldofmachinelearningandlookstodevelopevenmorerapidlyinthecomingyearsasadvancementsincomputerhardwareandstatisticalmethodologiescontinuetomaketheirwayintotheestablishedcanon. Trial Trynow:createMLprojectscode-free StartforFree Abouttheauthor DataRobot TheNextGenerationofAI DataRobotAICloudisthenextgenerationofAI.Theunifiedplatformisbuiltforalldatatypes,allusers,andallenvironmentstodelivercriticalbusinessinsightsforeveryorganization.DataRobotistrustedbyglobalcustomersacrossindustriesandverticals,includingathirdoftheFortune50.Formoreinformation,visithttps://www.datarobot.com/. MeetDataRobot Listentotheblog   Sharethispost SubscribetoDataRobotBlog SubscribeNow FirstName LastName Email CountryPleaseselect... 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Close RelatedPosts Seeotherpostsin AI&MLExpertise BuildBetterModelswithMLOps January13,2021 · 4min read Althoughstillarelativelynewfield,thepresentglobalsituationandchangeshavebroughtAI tothepointwheremachinelearningmodelsreallyneedtostartshowingtheirvalue.Toachievethat,DataRobotprovidesasolutionfororganizationstobuildanMLOpsfoundationthatallowsdata,development,andproductionteamstoworktogethertosuccessfullydeployandmanagemachinelearningservices… ReadMore   January13,2021 · 4min read AIAcrossIndustries DataRobotisAcquiringAlgorithmia,EnhancingLeadingMLOpsInfrastructuretoGetModelstoProductionFast,withOptimizedGPUWorkloadsatScale July27,2021 · 4min read Inaglobalmarketplacewheredecision-makingneedstohappenwithincreasingvelocity,datascienceteamsoftenneednotonlytospeeduptheirmodelingdeploymentbutalsodoitatscaleacrosstheirentireenterprise.Often,theyaredoingthiswithsmallerteamsinplacethantheyneedduetotheshortageofdatascientists.It’snowonder,then,that… ReadMore   July27,2021 · 4min read DataRobotRelease IntroducingtheMLOpsManagementAgent June16,2021 · 4min read Inseveralearlierblogposts,wehavefocusedonwhatweatDataRobotcalltheAIproductiongap,whichreferstothegapthatmakesitdifficulttotransitionmodelsfromthedatascienceteamswhodevelopthemtotheITandDevOpsteamswhoareresponsiblefordeployingandmonitoringtheminproduction.DeploymentandMonitoringChallengesOurDataRobot… ReadMore   June16,2021 · 4min read Seeotherpostsin SubscribetoourBlog FirstName LastName Email CountryPleaseselect... 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