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Course: LING 521, Fall 2009School: UPenn
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521 LING Phonetics Practicum Liberman & Yuan, Spring 2006 Lecture5 Probability Machinelearning (ataglance) Feb.8,2006 Probability Itisatruthvaluethatallowsyoutobeuncertain,ifyouwish. Itdescribeshowmuchyouknowaboutasituation. Itcountshowbigaspecifiedsubsetofpossibleresultsis, relativetothesizeoftheuniverse(thatmatters). Itisanestimateofhowoftensomethinghappensunder specifiedconditions....
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521 LING Phonetics Practicum
Liberman & Yuan, Spring 2006
Lecture5
Probability Machinelearning (ataglance) Feb.8,2006
Probability
Itisatruthvaluethatallowsyoutobeuncertain,ifyouwish. Itdescribeshowmuchyouknowaboutasituation. Itcountshowbigaspecifiedsubsetofpossibleresultsis, relativetothesizeoftheuniverse(thatmatters). Itisanestimateofhowoftensomethinghappensunder specifiedconditions. Probabilitiesarenotconcretethinksthatyoucantouch.They arealmostPaltonicideals,propertiesofamodelthatyoucan nevermeasuredirectlyorexactly,butcanonlyestimatefrom thedata.
2
[fromGregKochanski,OxfordUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
Cointossingexample
Inprobabilitytheoryanexperimentisarepeatableprocedure formakinganobservation Belowisonesuchexperiment: Tossingacoingivestwopossibleoutcomes:headsortails Assumingthatthecoinisfairthereisa50%chance (proportion=.5)thatanygiventosswillgiveahead Althoughtheproportionofheadswilllikelybequitevariable atfirst,aswetossalargenumberoftimesitwillapproach.5 Wecansay,then,thattheprobabilityoftossingaheadis.5
3
[fromBobAndersen,McMasterUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
Cointossingexample
4
[fromBobAndersen,McMasterUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
Counting
5
[fromRobMalouf,SanDiegoStateUniverisity]
LING 521 Phonetics Practicum, Liberman & Yuan
Counting
6
[fromRobMalouf,SanDiegoStateUniverisity]
LING 521 Phonetics Practicum, Liberman & Yuan
Probability
7
[fromRobMalouf,UniversityofGroningen]
LING 521 Phonetics Practicum, Liberman & Yuan
Therulesofprobability
1.Theprobabilityofaparticulareventoccurringisbetween0and1 [0P(event)1] 2.Allpossibleoutcomeswillsumto1[P(S)=1,whereSisthe samplespace. 3.Theprobabilitythataparticulareventdoesnotoccuris1minus theprobabilityoftheeventoccurring[P(eventdoesnot occur)=1P(event) 4.Iftwoeventshavenooutcomesincommon,theprobabilitythat oneoccursissimplythesumoftheindividualprobabilities[P(A orB)=P(A)+P(B).Knownastheadditionrulefordisjoint events. 5.IfAandBareindependent,theprobabilityofbothoccurringone aftertheotheristheproductoftheindividualprobabilities:[P(A andB)=P(A)P(B)]
[fromBobAndersen,McMasterUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
8
Therulesofprobability
6.Iftwoevents,AandB,arenotdisjointi.e.,theyhave outcomesincommonwecannotsimplyaddtheirindividual probabilitiestodeterminetheprobabilityoftheoutcomesofAor B Instead,P(AorB)=P(A)+P(B)P(AandB) Inotherwords,wemustnowsubtracttheoverlapping outcomes 7.Twoeventsaredependentiftheprobabilityofonedependson theother.TheconditionalprobabilityofaneventBgiventhe occurrenceofaneventAis:
9
P(B | A) =
P (A and B) P (A)
[fromBobAndersen,McMasterUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
Conditionalprobability
P(A|B)=FractionofworldsinwhichBistruethatalsohaveA true H=Haveaheadache F=ComingdownwithFlu P(H)=1/10 P(F)=1/40 P(H|F)=1/2 Headachesarerareandflu israrer,butifyoure comingdownwithflu theresa5050chance youllhaveaheadache.
[fromAndrewMoore,CarnegieMellonUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
10
Conditionalprobability
11
P(H|F)=Fractionoffluinflictedworldsinwhich youhaveaheadache #worldswithfluandheadache = #worldswithflu AreaofHandFregion = AreaofFregion P(H^F) = P(F)
H=Haveaheadache F=ComingdownwithFlu P(H)=1/10 P(F)=1/40 P(H|F)=1/2
[fromAndrewMoore,CarnegieMellonUniversity]
LING 521 Phonetics Practicum, & Liberman Yuan
ChainruleandBayesrule
Chainrule: P(A^B)=P(A|B)P(B) =P(B^A) =P(B|A)P(A) P(A^B^C^D)=P(A)P(B|A)P(C|A,B)P(D|A,B,C..)
12
Bayesrule: fromP(A,B)=P(A|B)P(B)=P(B|A)P(A)
P(A ^ B) P(B | A)P(A) P(A | B) = = P(B) P(B)
LING 521 Phonetics Practicum, Liberman & Yuan
Bayesianinference
S:stiffneck,M:meningitis P(S|M)=0.5,P(M)=1/50,000P(S)=1/20 Ihavestiffneck,shouldIworry?
13
P( S | M ) P( M ) P( M | S ) = P( S ) 0.5 1 / 50,000 = = 0.0002 1 / 20
LING 521 Phonetics Practicum, Liberman & Yuan
Machinelearning
14
[fromEthemAlpaydin,BogaziiUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
Machinelearning
15
[fromEthemAlpaydin,BogaziiUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
DecisionTrees
16
[fromEthemAlpaydin(adaptedfromAikChoonTan),BogaziiUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
DecisionTrees
17
[fromEthemAlpaydin(adaptedfromAikChoonTan),BogaziiUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
DecisionTrees
18
[fromEthemAlpaydin(adaptedfromAikChoonTan),BogaziiUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
DecisionTrees
Q1:Choosingthebestattributetosplitatatime:whatisthe measuretouse? Q2:Determiningwhentostopsplitting:avoidoverfitting Q3:Handlingcontinuousattributes:findthebreakpoint Q4:Handlingtrainingdatawithmissingattributevalues Q5:Handingattributeswithdifferentcosts
19
LING 521 Phonetics Practicum, Liberman & Yuan
Informationtheory:Entropy
20
[fromEthemAlpaydin(adaptedfromAikChoonTan),BogaziiUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
Informationtheory:Entropy
21
[fromEthemAlpaydin(adaptedfromAikChoonTan),BogaziiUniversity]
LING 521 Phonetics Practicum, Liberman & Yuan
Informationgain
Gain(S,A)=expectedreductioninentropyduetosortingonA.
22
| Sv | Gain( S , A) = Entropy ( S ) Entropy ( S v ) vValues ( A ) | S |
def
ChoosetheAwiththemaxinformationgain. (a.k.a.choosetheAwiththeminaverageentropy)
LING 521 Phonetics Practicum, Liberman & Yuan
Overfitting
23
[fromEthemAlpa...
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