Bayes Factors ffl Examples

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Lecture:

BayesianReasoning

KevinB.Korb

SchoolofComputerScience

andSoftwareEngineering

Of ce:G07/Building63

Of cehours:Mon4pm;Tue,Thr2pm

phone:9905-5198

email:korb@csse.monash.edu.au

HTTP://WWW.CSSE.MONASH.EDU.AU/ KORB

1Korb

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LectureOverview

BreastCancer

Probabilities

VennDiagrams

ConditionalProbability

Bayes’Theorem

Conditionalization

BettingandOdds

Odds-LikelihoodBayes

BayesFactors

Examples

2Korb

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KorbBreastCancer

Ascreeningtesthasa90%chanceofregisteringbreast

cancerifitexists,aswellasa20%chanceoffalsely

registeringcancerwhenitdoesnotexist.Aboutoneinone

hundredwomenrequestingthescreeningtestendup

diagnosedwithbreastcancer.Ms.Xhasjustbeentoldthat

herscreeningtestwaspositive.Whatistheprobabilitythatshehasbreastcancer?

Whatistheprobabilityofbreastcancer?

1.About1%.

2.About90%.

3.About85%.

4.About80%.

5.About4%.

3

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Probabilities

Whatareprobabilities?

Usedtomeasureuncertainty

Relatedtofrequencies

–FrequencyInterpretation

JohnVenn,RichardvonMises

Relatedtorationalbettingodds

–BayesianInterpretation

FrankRamsey,BrunodeFinetti

“Whocares?”Interpretation

–Kolmogorov’saxioms

Whatevertheinterpretation,dealingwith

probability(uncertainty)isakeyreasoning

ability.

4Korb

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Figure3:Venn

DiagramsFigure1:

Figure2:

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Conditional

Probability

Figure4:

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Bayes’Theorem

—RevThomasBayes(1764[posthumously])

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Conditionalization

Conditionalizingontheevidence:

Given

–

–TotalEvidence:andonlyislearned

Adoptasyourposteriorprobability

–

—i.e.,applyBayes’theorem

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ConditionalizationExample

Breastcancer:wearegiven

and

Whatis?

ByBayes’theoremArewehavingfunyet?

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BettingandOdds

Given,whatisafairbeton?

)ifandonlyiftheAbetisfair(relativeto

expectedvalueofthebetiszero.

I.e.,

Thiswillbesatis edincasebettingoddsareset:

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BettingandOdds

Example:

Afairbetthata(fair)diewillshowanumber

requiresoddsof

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OddsandProbabilities

Asnoted,So,oddsandprobabilitiesareinterchangeable.

Likewise,posterioroddsandposteriorprobabilities:Thisgivesusanewwayofconditionalizing.

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Odds-LikelihoodBayes

Odds-LikelihoodBayes’Theorem:

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BreastCancerExample

Breastcancer:wearegiven

and

Whatis?

Algorithm:

1.Find

2.Find

3.Multiply(i.e.,get)

So:

1.

2.

3.

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BreastCancerExample

Intuitively,isafairlyweaktest;is

isabout4timeslarger,italowprior.Although

is4timesasmallnumber.

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DNAExample

ADNAmatchbetweenthedefendantandacrimescenebloodsamplehasaprobabilityof1/100000ifthedefendantis

innocent.Thereisnoothersigni cantevidence.Whatistheprobabilityofguilt?

Solution.Thegiven1/100000isjust,whereisHowever,theDNAnoevidencepriorisgiven,andsoassertsnoposteriorinnocence.canbecalculated.

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DNAExample

ADNAmatchbetweenthedefendantandacrimescenebloodsamplehasaprobabilityof1/100000ifthedefendantis

innocent.Thereisnoothersigni cantevidence.Supposeweagreethatthepriorprobabilityofguiltunderthe

(unspeci ed)circumstancesis10%.Whatistheprobabilityofguilt?

Solution..andwecantake(theDNAmatchassumingguilt)asabout1;so.Thus,.I.e.,theoddsofguiltareverystrong.

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DNAExample

ADNAmatchbetweenthedefendantandacrimescenebloodsamplehasaprobabilityof1/100000ifthedefendantis

innocent.Thereisnoothersigni cantevidence.ThesuspecthasbeenpickedupthroughauniversalscreeningprogramappliedtoallAustraliansseekingaMedicarecard.Sofar,10millionpeoplehavebeenscreened.Whatistheprobabilityofguilt?

Solution..

andwecantake(theDNAmatchassumingguilt)asabout1;so.Thus,

oddsofguiltareveryweak..I.e.,the

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References

Someusefulintroductoryandphilosophicaltreatmentsofprobabilisticreasoning.

Baird,D.(1992)Inductivelogic:Probabilityandstatistics.

PrenticeHall.

Bayes,T.(1764).Anessaytowardssolvingaprobleminthe

doctrineofchances.PhilTransoftheRoyalSocof

London.ReprintedinBiometrika,45(1958),296-315.Good,I.J.(1983).Goodthinking.UnivofMinnesota.

Howson,C.andUrbach,P.(1992)Scienti creasoning:The

Bayesianapproach,secondedition.OpenCourt.

Jeffrey,R.().Thelogicofdecision,secondedition.

McGraw-Hill.

Skyrms,B.(1986).Choiceandchance,thirdedition.Wadsworth.

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