CSC412/2506Spring2017Probabilis7cGraphicalModels

Lecture3:DirectedGraphicalModels

andLatentVariables

BasedonslidesbyRichardZemel

Learningoutcomes

•  Whataspectsofamodelcanweexpressusinggraphicalnotation?

•  Whichaspectsarenotcapturedinthisway?•  Howdoindependencieschangeasaresultofconditioning?

•  Reasonsforusinglatentvariables•  Commonmotifssuchasmixturesandchains•  Howtointegrateoutunobservedvariables

JointProbabili7es

•  Chainruleimpliesthatanyjointdistributionequals

•  Directedgraphicalmodelimpliesarestrictedfactorization

p(x1:D) = p(x1)p(x2|x1)p(x3|x1, x2)p(x4|x1, x2, x3)...p(xD|x1:D�1)

Condi7onalIndependence•  Notation:xA⊥xB|xC

•  DeGinition:two(setsof)variablesxAandxBareconditionallyindependentgivenathirdxCif:

whichisequivalenttosaying

•  Onlyasubsetofalldistributionsrespectanygiven(nontrivial)conditionalindependencestatement.ThesubsetofdistributionsthatrespectalltheCIassumptionswemakeisthefamilyofdistributionsconsistentwithourassumptions.

•  Probabilisticgraphicalmodelsareapowerful,elegantandsimplewaytospecifysuchafamily.

P (xA,xB |xC) = P (xA|xC)P (xB |xC) ; 8xC

P (xA|xB ,xC) = P (xA|xC) ; 8xC

DirectedGraphicalModels•  Considerdirectedacyclicgraphsovernvariables.•  Eachnodehas(possiblyempty)setofparentsπi•  Wecanthenwrite

•  Hencewefactorizethejointintermsoflocalconditionalprobabilities

•  Exponentialin“fan-in”ofeachnodeinsteadofinN

P (x1, ...,xN ) =Y

i

P (xi|x⇡i)

Condi7onalIndependenceinDAGs

•  Ifweorderthenodesinadirectedgraphicalmodelsothatparentsalwayscomebeforetheirchildrenintheorderingthenthegraphicalmodelimpliesthefollowingaboutthedistribution:

wherearethenodescomingbeforexithatarenotitsparents

•  Inotherwords,theDAGistellingusthateachvariableisconditionallyindependentofitsnon-descendantsgivenitsparents.

•  Suchanorderingiscalleda“topological”ordering

x⇡i

{xi ? x⇡i |x⇡i} ; 8i

ExampleDAGConsiderthissixnodenetwork:Thejointprobabilityisnow:

MissingEdges•  Keypointaboutdirectedgraphicalmodels:

Missingedgesimplyconditionalindependence•  Rememberthatbythechainrulewecanalwayswritethefulljointasaproductofconditionals,givenanordering:

•  IfthejointisrepresentedbyaDAGM,thensomeoftheconditionedvariablesontherighthandsidesaremissing.

•  Thisisequivalenttoenforcingconditionalindependence.•  Startwiththe“idiot’sgraph”:eachnodehasallpreviousnodesintheorderingasitsparents.

•  NowremoveedgestogetyourDAG.•  Removinganedgeintonodeieliminatesanargumentfromtheconditionalprobabilityfactor

P (x1,x2, ...) = P (x1)P (x2|x1)P (x3|x2,x1)P (x4|x3,x2,x1)

P (xi|x1,x2, ...,xi�1)

D-Separa7on•  D-separation,ordirected-separationisanotionofconnectednessinDAGMsinwhichtwo(setsof)variablesmayormaynotbeconnectedconditionedonathird(setof)variable.

•  D-connectionimpliesconditionaldependenceandd-separationimpliesconditionalindependence.

•  Inparticular,wesaythatxA⊥xB|xCifeveryvariableinAisd-separatedfromeveryvariableinBconditionedonallthevariablesinC.

•  Tocheckifanindependenceistrue,wecancyclethrougheachnodeinA,doadepth-GirstsearchtoreacheverynodeinB,andexaminethepathbetweenthem.Ifallofthepathsared-separated,thenwecanassertxA⊥xB|xC

•  Thus,itwillbesufGicienttoconsidertriplesofnodes.(Why?)•  Pictorially,whenweconditiononanode,weshadeitin.

Chain

•  Q:Whenweconditionony,arexandzindependent?

whichimpliesandthereforex⊥z|y•  Thinkofxasthepast,yasthepresentandzasthefuture.

P (x,y, z) = P (x)P (y|x)P (z|y)

P (z|x,y) = P (x,y, z)

P (x,y)

=P (x)P (y|x)P (z|y)

P (x)P (y|x)= P (z|y)

CommonCause

•  Q:Whenweconditionony,arexandzindependent?

whichimpliesandthereforex⊥z|y

= P (x|y)P (z|y)

=P (y)P (x|y)P (z|y)

P (y)

P (x, z|y) = P (x,y, z)

P (y)

P (x,y, z) = P (y)P (x|y)P (z|y)

ExplainingAway

•  Q:Whenweconditionony,arexandzindependent?•  xandzaremarginallyindependent,butgivenytheyareconditionallydependent.

•  Thisimportanteffectiscalledexplainingaway(Berkson’sparadox.)

•  Forexample,Gliptwocoinsindependently;letx=coin1,z=coin2.•  Lety=1ifthecoinscomeupthesameandy=0ifdifferent.•  xandzareindependent,butifItellyouy,theybecomecoupled!

P (x,y, z) = P (x)P (z)P (y|x, z)

Bayes-BallAlgorithm•  TocheckifxA⊥xB|xCweneedtocheckifeveryvariableinAisd-separatedfromeveryvariableinBconditionedonallvarsinC.

•  Inotherwords,giventhatallthenodesinxCareclamped,whenwewigglenodesxAcanwechangeanyofthenodesinxB?

•  TheBayes-BallAlgorithmisasuchad-separationtest.•  WeshadeallnodesxC,placeballsateachnodeinxA(orxB),letthembouncearoundaccordingtosomerules,andthenaskifanyoftheballsreachanyofthenodesinxB(orxA).

Bayes-BallRules•  Thethreecasesweconsideredtellusrules:

Bayes-BallBoundaryRules

•  Wealsoneedtheboundaryconditions:

•  Here’satrickfortheexplainingawaycase:Ifyoranyofitsdescendantsisshaded,theballpassesthrough.

•  Noticeballscantraveloppositetoedgedirections.

CanonicalMicrographs

ExamplesofBayes-BallAlgorithm

ExamplesofBayes-BallAlgorithm

•  Notice:ballscantraveloppositetoedgedirection

Plates

Plates&Parameters

•  SinceBayesianmethodstreatparametersasrandomvariables,wewouldliketoincludetheminthegraphicalmodel.

•  Onewaytodothisistorepeatalltheiidobservationsexplicitlyandshowtheparameteronlyonce.

•  Abetterwayistouseplates,inwhichrepeatedquantitiesthatareiidareputinabox.

Plates:MacrosforRepeatedStructures•  Platesarelike“macros”thatallowyoutodrawaverycomplicatedgraphicalmodelwithasimplernotation.

•  Therulesofplatesaresimple:repeateverystructureinaboxanumberoftimesgivenbytheintegerinthecornerofthebox(e.g.N),updatingtheplateindexvariable(e.g.n)asyougo.

•  Duplicateeveryarrowgoingintotheplateandeveryarrowleavingtheplatebyconnectingthearrowstoeachcopyofthestructure.

Nested/Intersec7ngPlates

•  Platescanbenested,inwhichcasetheirarrowsgetduplicatedalso,accordingtotherule:drawanarrowfromeverycopyofthesourcenodetoeverycopyofthedestinationnode.

•  Platescanalsocross(intersect),inwhichcasethenodesattheintersectionhavemultipleindicesandgetduplicatedanumberoftimesequaltotheproductoftheduplicationnumbersonalltheplatescontainingthem.

Example:NestedPlates

ExampleDAGM:MarkovChain

•  MarkovProperty:Conditionedonthepresent,thepastandfutureareindependent

UnobservedVariables•  CertainvariablesQinourmodelsmaybeunobserved,eithersomeofthetimeoralways,eitherattrainingtimeorattesttime

•  Graphically,wewilluseshadingtoindicateobservation

Par7allyUnobserved(Missing)Variables•  Ifvariablesareoccasionallyunobservedtheyaremissingdata,e.g.,undeGinedinputs,missingclasslabels,erroneoustargetvalues

•  Inthiscase,wecanstillmodelthejointdistribution,butwedeGineanewcostfunctioninwhichwesumoutormarginalizethemissingvaluesattrainingortesttime:

Recallthat

`(✓;D) =

X

complete

log p(xc,yc|✓) +X

missing

log p(xm|✓)

=

X

complete

log p(xc,yc|✓) +X

missing

log

X

y

p(xm,y|✓)

p(x) =X

q

p(x, q)

LatentVariables•  Whattodowhenavariablezisalwaysunobserved?Dependsonwhereitappearsinourmodel.Ifweneverconditiononitwhencomputingtheprobabilityofthevariableswedoobserve,thenwecanjustforgetaboutitandintegrateitout.e.g.,giveny,xGitthemodelp(z,y|x)=p(z|y)p(y|x,w)p(w).(Inotherwordsifitisaleafnode.)

•  Butifzisconditionedon,weneedtom odelit:odelit:

e.g.giveny,xGitthemodelp(y|x)=Σzp(y|x,z)p(z)

WhereDoLatentVariablesComeFrom?•  Latentvariablesmayappearnaturally,fromthestructureoftheproblem,becausesomethingwasn’tmeasured,becauseoffaultysensors,occlusion,privacy,etc.

•  Butalso,wemaywanttointentionallyintroducelatentvariablestomodelcomplexdependenciesbetweenvariableswithoutlookingatthedependenciesbetweenthemdirectly.Thiscanactuallysimplifythemodel(e.g.,mixtures).

LatentVariablesModels&Regression

•  YoucanthinkofclusteringastheproblemofclassiGicationwithmissingclasslabels.

•  Youcanthinkoffactormodels(suchasfactoranalysis,PCA,ICA,etc.)aslinearornonlinearregressionwithmissinginputs.

WhyisLearningHarder?

•  Infullyobservediidsettings,theprobabilitymodelisaproduct,thustheloglikelihoodisasumwheretermsdecouple.(Atleastfordirectedmodels.)

•  Withlatentvariables,theprobabilityalreadycontainsasu m,sotheloglikelihoodhasallparameterscoupledtogetherviaz:rscoupledtogetherviaz:

(Justaswiththepartitionfunctioninundirectedmodels)

`(✓;D) = log p(x, z|✓)

= log p(z|✓z

) + log p(x|z, ✓x

)

`(✓;D) = log

X

z

p(x, z|✓)

= log

X

z

p(z|✓z

) + log p(x|z, ✓x

)

WhyisLearningHarder?

•  Likelihood couplesparameters:

•  Wecantreatthisasablackboxprobabilityfunctionandjusttrytooptimizethelikelihoodasafunctionofθ(e.g.gradientdescent).However,sometimestakingadvantageofthelatentvariablestructurecanmakeparameterestimationeasier.

•  Goodnews:soonwewillseehowtodealwithlatentvariables.•  Basictrick:putatractabledistributiononthevaluesyoudon’tknow.Basicmath:useconvexitytolowerboundthelikelihood.

= log

X

z

p(z|✓z

) + log p(x|z, ✓x

)

MixtureModels•  Mostbasiclatentvariablemodelwithasinglediscretenodez.•  Allowsdifferentsubmodels(experts)tocontributetothe(conditional)densitymodelindifferentpartsofthespace.

•  Divide&conqueridea:usesimplepartstobuildcomplexmodels(e.g.,multimodaldensities,orpiecewise-linearregressions).

MixtureDensi7es•  ExactlylikeaclassiGicationmodelbuttheclassisunobservedandsowesumitout.Whatwegetisaperfectlyvaliddensity:

wherethe“mixingproportions”addtoone:Σkαk=1.•  WecanuseBayes’ruletocomputetheposteriorprobabilityofthemixturecomponentgivensomedata:

thesequantitiesarecalledresponsibilities.

p(z = k|x, ✓) = ↵kpk(x|✓k)Pj ↵jpj(x|✓j)

p(x|✓) =KX

k=1

p(z = k|✓z)p(x|z = k, ✓k)

=X

k

↵kpk(x|✓k)

Example:GaussianMixtureModels•  ConsideramixtureofKGaussiancomponents:

•  Densitymodel:p(x|θ)isafamiliaritysignal.Clustering:p(z|x,θ)istheassignmentrule,−l(θ)isthecost.

p(x|✓) =X

k

↵kN (x|µk,⌃k)

p(z = k|x, ✓) = ↵kN (x|µk,⌃k)Pj ↵jN (x|µj ,⌃j)

`(✓;D) =

X

n

log

X

k

↵kN (x

(n)|µk,⌃k)

Example:MixturesofExperts•  Alsocalledconditionalmixtures.Exactlylikeaclass-conditionalmodelbuttheclassisunobservedandsowesumitoutagain:

where

•  Harder:mustlearnα(x)(unlesschosezindependentofx).•  WecanstilluseBayes’ruletocomputetheposteriorprobabilityofthemixturecomponentgivensomedata:

thisfunctionisoftencalledthegatingfunction.

p(y|x, ✓) =KX

k=1

p(z = k|x, ✓z)p(y|z = k,x, ✓k)

=X

k

↵k(x|✓z)pk(y|x, ✓k)X

k

↵k(x) = 1 8x

p(z = k|x,y, ✓) = ↵k(x)pk(y|x, ✓k)Pj ↵j(x)pj(y|x, ✓j)

Example:MixturesofLinearRegressionExperts•  EachexpertgeneratesdataaccordingtoalinearfunctionoftheinputplusadditiveGaussian noise:

•  The“gate”functioncanbeasoftmaxclassiGicationmachine

•  Remember:wearenotmodelingthedensityoftheinputsx

p(y|x, ✓) =X

k

↵kN (y|�Tk x,�

2k)

↵k(x) = p(z = k|x) = e⌘Tk x

Pj e

⌘Tj x

GradientLearningwithMixtures

•  Wecanlearnmixturedensitiesusinggradientdescentonthelikelihoodasusual.Thegradientsarequiteinteresting:

•  Inotherwords,thegradientistheresponsibilityweightedsumoftheindividualloglikelihoodgradients

`(✓) = log p(x|✓) = log

X

k

↵kpk(x|✓k)

@`

@✓=

1

p(x|✓)X

k

↵k@pk(x|✓k)

@✓

=

X

k

↵k1

p(x|✓)pk(x|✓k)@ log pk(x|✓k)

@✓

=X

k

↵kpk(x|✓k)p(x|✓)

@`k@✓k

=X

k

↵krk@`k@✓k

ParameterConstraints

•  Ifwewanttousegeneraloptimizations(e.g.,conjugategradient)tolearnlatentvariablemodels,weoftenhavetomakesureparametersrespectcertainconstraints(e.g.,Σkαk=1,ΣkαkpositivedeGinite)

•  Agoodtrickistore-parameterizethesequantitiesintermsofunconstrainedvalues.Formixingproportions,usethesoftmax:

•  Forcovariancematrices,usetheCholeskydecomposition

whereAisupperdiagonalwithpositivediagonal

↵k =

exp(qk)Pj exp(qj)

⌃�1 = ATA |⌃|�1/2 =Y

i

Aii

Aii = exp(ri) > 0 Aij = aij (j > i) Aij = 0 (j < i)

Logsumexp•  Oftenyoucaneasilycomputebk=logp(x|z=k,θk),

butitwillbeverynegative,say-106orsmaller.•  Now,tocomputel=logp(x|θ)youneedtocompute(e.g.,forcalculatingresponsibilitiesattesttimeorforlearning)

•  Careful!Donotcomputethisbydoinglog(sum(exp(b))).YouwillgetunderGlowandanincorrectanswer.

•  Insteaddothis:–AddaconstantexponentBtoallthevaluesbksuchthatthe

largestvalueequalszero:B=max(b).–Computelog(sum(exp(b-B)))+B.•  Example:iflogp(x|z=1)=−120andlogp(x|z=2)=−120,whatislogp(x)=log[p(x|z=1)+p(x|z=2)]?Answer:log[2e−120]=−120+log2.

•  Ruleofthumb:neveruselogorexpbyitself

log

Pk e

bk

HiddenMarkovModels(HMMs)

•  Averypopularformoflatentvariablemodel

•  ZtàHiddenstatestakingoneofKdiscretevalues•  XtàObservationstakingvaluesinanyspace

Example:discrete,MobservationsymbolsB 2 <KxM

p(xt = j|zt = k) = Bkj

InferenceinGraphicalModels

xEàObservedevidencevariables(subsetofnodes)xFàunobservedquerynodeswe’dliketoinferxRàremainingvariables,extraneoustothisquerybutpartofthegivengraphicalrepresentation

InferencewithTwoVariables

Tablelook-up

Bayes’Rulep(x|y = y) =

p(y|x)p(x)p(y)

p(y|x = x)

NaïveInference

•  Supposeeachvariabletakesoneofkdiscretevalues

•  CostsO(k)operationstoupdateeachofO(k5)tableentries•  Usefactorizationanddistributedlawtoreducecomplexity

p(x1, x2, ..., x5) =X

x6

p(x1)p(x2|x1)p(x3|x1)p(x4|x2)p(x5|x3)p(x6|x2, x5)

= p(x1)p(x2|x1)p(x3|x1)p(x4|x2)p(x5|x3)X

x6

p(x6|x2, x5)

InferenceinDirectedGraphs

InferenceinDirectedGraphs

InferenceinDirectedGraphs

Learningoutcomes

•  Whataspectsofamodelcanweexpressusinggraphicalnotation?

•  Whichaspectsarenotcapturedinthisway?•  Howdoindependencieschangeasaresultofconditioning?

•  Reasonsforusinglatentvariables•  Commonmotifssuchasmixturesandchains•  Howtointegrateoutunobservedvariables

Ques7ons?

•  Thursday:Tutorialonautomaticdifferentiation•  Thisweek:Assignment1released