CSE473:Ar+ficialIntelligence

Probability

Instructors:LukeZe?lemoyer---UniversityofWashington[TheseslideswerecreatedbyDanKleinandPieterAbbeelforCS188IntrotoAIatUCBerkeley.AllCS188materialsareavailableath?p://ai.berkeley.edu.]

Today

§  Probability§  RandomVariables§  JointandMarginalDistribu+ons§  Condi+onalDistribu+on§  ProductRule,ChainRule,Bayes’Rule§  Inference§  Independence

§  You’llneedallthisstuffALOTforthenextfewweeks,somakesureyougooveritnow!

InferenceinGhostbusters

§  Aghostisinthegridsomewhere

§  Sensorreadingstellhowcloseasquareistotheghost§  Ontheghost:red§  1or2away:orange§  3or4away:yellow§  5+away:green

P(red|3) P(orange|3) P(yellow|3) P(green|3)0.05 0.15 0.5 0.3

§ Sensorsarenoisy,butweknowP(Color|Distance)

[Demo:Ghostbuster–noprobability(L12D1)]

Uncertainty

§  Generalsitua+on:

§  Observedvariables(evidence):Agentknowscertainthingsaboutthestateoftheworld(e.g.,sensorreadingsorsymptoms)

§  Unobservedvariables:Agentneedstoreasonaboutotheraspects(e.g.whereanobjectisorwhatdiseaseispresent)

§  Model:Agentknowssomethingabouthowtheknownvariablesrelatetotheunknownvariables

§  Probabilis+creasoninggivesusaframeworkformanagingourbeliefsandknowledge

RandomVariables

§  Arandomvariableissomeaspectoftheworldaboutwhichwe(may)haveuncertainty

§  R=Isitraining?§  T=Isithotorcold?§  D=Howlongwillittaketodrivetowork?§  L=Whereistheghost?

§  Wedenoterandomvariableswithcapitalle?ers

§  Randomvariableshavedomains§  Rin{true,false}(ooenwriteas{+r,-r})§  Tin{hot,cold}§  Din[0,∞)§  Linpossibleloca+ons,maybe{(0,0),(0,1),…}

ProbabilityDistribu+ons

§  Associateaprobabilitywitheachvalue

§  Temperature:

T P

hot 0.5

cold 0.5

W P

sun 0.6

rain 0.1

fog 0.3

meteor 0.0

§  Weather:

Shorthandnota+on:

OKifalldomainentriesareunique

ProbabilityDistribu+ons

§  Unobservedrandomvariableshavedistribu+ons

§  Adistribu+onisaTABLEofprobabili+esofvalues

§  Aprobability(lowercasevalue)isasinglenumber

§  Musthave:and

T P

hot 0.5

cold 0.5

W P

sun 0.6

rain 0.1

fog 0.3

meteor 0.0

JointDistribu+ons§  Ajointdistribu-onoverasetofrandomvariables:specifiesarealnumberforeachassignment(oroutcome):

§  Mustobey:

§  Sizeofdistribu+onifnvariableswithdomainsizesd?

§  Forallbutthesmallestdistribu+ons,imprac+caltowriteout!

T W Phot sun 0.4hot rain 0.1cold sun 0.2cold rain 0.3

Probabilis+cModels

§  Aprobabilis+cmodelisajointdistribu+onoverasetofrandomvariables

§  Probabilis+cmodels:§  (Random)variableswithdomains§  Assignmentsarecalledoutcomes§  Jointdistribu+ons:saywhetherassignments(outcomes)arelikely

§  Normalized:sumto1.0§  Ideally:onlycertainvariablesdirectlyinteract

T W Phot sun 0.4hot rain 0.1cold sun 0.2cold rain 0.3

Distribu+onoverT,W

Events§  AneventisasetEofoutcomes

§  Fromajointdistribu+on,wecancalculatetheprobabilityofanyevent

§  Probabilitythatit’shotANDsunny?

§  Probabilitythatit’shot?

§  Probabilitythatit’shotORsunny?

§  Typically,theeventswecareaboutarepar-alassignments,likeP(T=hot)

T W Phot sun 0.4hot rain 0.1cold sun 0.2cold rain 0.3

Quiz:Events

§  P(+x,+y)?

§  P(+x)?

§  P(-yOR+x)?

X Y P+x +y 0.2+x -y 0.3-x +y 0.4-x -y 0.1

=0.2

0.2+0.3 = 0.5

0.2+0.3+0.1 = 0.6

MarginalDistribu+ons

§  Marginaldistribu+onsaresub-tableswhicheliminatevariables§  Marginaliza+on(summingout):Combinecollapsedrowsbyadding

T W Phot sun 0.4hot rain 0.1cold sun 0.2cold rain 0.3

T Phot 0.5cold 0.5

W Psun 0.6rain 0.4

Quiz:MarginalDistribu+ons

X Y P+x +y 0.2+x -y 0.3-x +y 0.4-x -y 0.1

X P+x-x

Y P+y-y

0.5

0.5

0.6

0.4

Condi+onalProbabili+es§  Asimplerela+onbetweenjointandcondi+onalprobabili+es

§  Infact,thisistakenasthedefini-onofacondi+onalprobability

T W Phot sun 0.4hot rain 0.1cold sun 0.2cold rain 0.3

P(b)P(a)

P(a,b)

Quiz:Condi+onalProbabili+es

X Y P+x +y 0.2+x -y 0.3-x +y 0.4-x -y 0.1

§  P(+x|+y)?

§  P(-x|+y)?

§  P(-y|+x)?

0.2 / (0.2+0.4) = 1/3

0.4 / (0.2+0.4) = 2/3

0.3 / (0.2+0.3) = 3/5

Condi+onalDistribu+ons

§  Condi+onaldistribu+onsareprobabilitydistribu+onsoversomevariablesgivenfixedvaluesofothers

T W Phot sun 0.4hot rain 0.1cold sun 0.2cold rain 0.3

W Psun 0.8rain 0.2

W Psun 0.4rain 0.6

Condi+onalDistribu+ons JointDistribu+on

Probabilis+cInference

§  Probabilis+cinference:computeadesiredprobabilityfromotherknownprobabili+es(e.g.condi+onalfromjoint)

§  Wegenerallycomputecondi+onalprobabili+es§  P(on+me|noreportedaccidents)=0.90§  Theserepresenttheagent’sbeliefsgiventheevidence

§  Probabili+eschangewithnewevidence:§  P(on+me|noaccidents,5a.m.)=0.95§  P(on+me|noaccidents,5a.m.,raining)=0.80§  Observingnewevidencecausesbeliefstobeupdated

InferencebyEnumera+on§  Generalcase:

§  Evidencevariables:§  Query*variable:§  Hiddenvariables: Allvariables

*Worksfinewithmul-plequeryvariables,too

§  Wewant:

§  Step1:Selecttheentriesconsistentwiththeevidence

§  Step2:SumoutHtogetjointofQueryandevidence

§  Step3:Normalize

⇥ 1

Z

InferencebyEnumera+on

§  P(W)?

§  P(W|winter)?

§  P(W|winter,hot)?

S T W Psummer hot sun 0.30summer hot rain 0.05summer cold sun 0.10summer cold rain 0.05winter hot sun 0.10winter hot rain 0.05winter cold sun 0.15winter cold rain 0.20

W Psun 0.65rain 0.35

W Psun 0.25rain 0.25 Z = 0.5

Normalize W Psun 0.5rain 0.5

W Psun 0.1rain 0.05 Z = 0.15

Normalize W Psun 0.66rain 0.33

§  Obviousproblems:

§  Worst-case+mecomplexityO(dn)

§  SpacecomplexityO(dn)tostorethejointdistribu+on

InferencebyEnumera+on

TheProductRule

§  Some+meshavecondi+onaldistribu+onsbutwantthejoint

TheProductRule

§  Example:

R P

sun 0.8

rain 0.2

D W P

wet sun 0.1

dry sun 0.9

wet rain 0.7

dry rain 0.3

D W P

wet sun 0.08

dry sun 0.72

wet rain 0.14

dry rain 0.06

TheChainRule

§  Moregenerally,canalwayswriteanyjointdistribu+onasanincrementalproductofcondi+onaldistribu+ons

§  Whyisthisalwaystrue?

BayesRule

Bayes’Rule

§  Twowaystofactorajointdistribu+onovertwovariables:

§  Dividing,weget:

§  Whyisthisatallhelpful?

§  Letsusbuildonecondi+onalfromitsreverse§  Ooenonecondi+onalistrickybuttheotheroneissimple§  Founda+onofmanysystemswe’llseelater(e.g.ASR,MT)

§  IntherunningformostimportantAIequa+on!

That’smyrule!

InferencewithBayes’Rule

§  Example:Diagnos+cprobabilityfromcausalprobability:

§  Example:§  M:meningi+s,S:s+ffneck

§  Note:posteriorprobabilityofmeningi+ss+llverysmall§  Note:youshoulds+llgets+ffneckscheckedout!Why?

Examplegivens

P (+s|�m) = 0.01

P (+m|+ s) =P (+s|+m)P (+m)

P (+s)=

P (+s|+m)P (+m)

P (+s|+m)P (+m) + P (+s|�m)P (�m)=

0.8⇥ 0.0001

0.8⇥ 0.0001 + 0.01⇥ 0.9999= 0.007937

P (+m) = 0.0001P (+s|+m) = 0.8

P (cause|e↵ect) = P (e↵ect|cause)P (cause)

P (e↵ect)

Quiz:Bayes’Rule

§  Given:

§ WhatisP(W|dry)?

R P

sun 0.8

rain 0.2

D W P

wet sun 0.1

dry sun 0.9

wet rain 0.7

dry rain 0.3

Ghostbusters,Revisited

§  Let’ssaywehavetwodistribu+ons:§  Priordistribu+onoverghostloca+on:P(G)

§  Let’ssaythisisuniform§  Sensorreadingmodel:P(R|G)

§  Given:weknowwhatoursensorsdo§  R=readingcolormeasuredat(1,1)§  E.g.P(R=yellow|G=(1,1))=0.1

§  Wecancalculatetheposteriordistribu+onP(G|r)overghostloca+onsgivenareadingusingBayes’rule:

[Demo:Ghostbuster–withprobability(L12D2)]

Independence

§  Twovariablesareindependentinajointdistribu+onif:

§  Saysthejointdistribu+onfactorsintoaproductoftwosimpleones§  Usuallyvariablesaren’tindependent!

§  Canuseindependenceasamodelingassump-on§  Independencecanbeasimplifyingassump+on§  Empiricaljointdistribu+ons:atbest“close”toindependent§  Whatcouldweassumefor{Weather,Traffic,Cavity}?

Example:Independence?

T W P

hot sun 0.4

hot rain 0.1

cold sun 0.2

cold rain 0.3

T W P

hot sun 0.3

hot rain 0.2

cold sun 0.3

cold rain 0.2

T P

hot 0.5

cold 0.5

W P

sun 0.6

rain 0.4

P2(T,W ) = P (T )P (W )

Example:Independence

§  Nfair,independentcoinflips:

H 0.5

T 0.5

H 0.5

T 0.5

H 0.5

T 0.5

Condi+onalIndependence

Condi+onalIndependence§  P(Toothache,Cavity,Catch)

§  IfIhaveacavity,theprobabilitythattheprobecatchesinitdoesn'tdependonwhetherIhaveatoothache:§  P(+catch|+toothache,+cavity)=P(+catch|+cavity)

§  ThesameindependenceholdsifIdon�thaveacavity:§  P(+catch|+toothache,-cavity)=P(+catch|-cavity)

§  Catchiscondi-onallyindependentofToothachegivenCavity:§  P(Catch|Toothache,Cavity)=P(Catch|Cavity)

§  Equivalentstatements:§  P(Toothache|Catch,Cavity)=P(Toothache|Cavity)§  P(Toothache,Catch|Cavity)=P(Toothache|Cavity)P(Catch|Cavity)§  Onecanbederivedfromtheothereasily

Condi+onalIndependence

§  Uncondi+onal(absolute)independenceveryrare(why?)

§  Condi-onalindependenceisourmostbasicandrobustformofknowledgeaboutuncertainenvironments.

§  Xiscondi+onallyindependentofYgivenZ

ifandonlyif:or,equivalently,ifandonlyif

Condi+onalIndependence

§  Whataboutthisdomain:

§  Traffic§  Umbrella§  Raining

Condi+onalIndependence

§  Whataboutthisdomain:

§  Fire§  Smoke§  Alarm

ProbabilityRecap

§  Condi+onalprobability

§  Productrule

§  Chainrule

§  X,Yindependentifandonlyif:

§  XandYarecondi+onallyindependentgivenZifandonlyif:

NextTime:MarkovModels