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CSCI 5582 Fall 2006

CSCI 5582

Artificial IntelligenceLecture 14

Jim Martin

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Today 10/1

! "e#ie$ %a&ic&

! More on inde'endence

! (rea)

! (aye&ian (elief *et&

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"e#ie$

! Joint +i&tri%ution&

! Atomic ,#ent&

! Inde'endence a&&um'tion&

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"e#ie$- Joint +i&tri%ution

Toot.ac.eTrue Toot.ac.eal&e

Ca#ity True 004 00

Ca#ity al&e 001 083

!,ac. cell re're&ent& a conunction of t.e #aria%le& in

t.e model

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Atomic ,#ent&

! T.e entrie& in t.e ta%le re're&entt.e 'ro%a%ilitie& of atomic e#ent&

,#ent& $.ere t.e #alue& of all t.e#aria%le& are &'ecified

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Inde'endence

! T$o #aria%le& A and ( areinde'endent iff 67A(9 67A9 In

ot.er $ord&: )no$ing ( gi#e& you noinformation a%out (

! ;r 67A

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Mental ,=erci&e

! >it. a fair coin $.ic. of t.efollo$ing t$o &e?uence& i& more

li)ely@ TTTTT

TTTTT

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Conditional Inde'endence

! Con&ider t.e denti&t 'ro%lem $it. B#aria%le&- ca#ity: toot.ac.e: catc.

! If I .a#e a ca#ity: t.en t.e c.ance&t.at t.ere $ill %e a catc. i&inde'endent of $.et.er or not I .a#e

a toot.ac.e a& $ell Ie 67Catc.Ca#ity

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Conditional Inde'endence

! "emem%er t.at .a#ing t.e ointdi&tri%ution o#er * #aria%le& allo$&

you to an&$er all t.e ?ue&tion&in#ol#ing t.o&e #aria%le&

! ,='loiting conditional inde'endence

allo$& u& to re're&ent t.e com'leteoint di&tri%ution $it. fe$er entrie&

Ie e$er t.an t.e 2*normally needed

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Conditional Inde'endence

! 67Ca#ity:Catc.:Toot.ac.e9

67Ca#ity967Catc.:Toot.ac.eCa#ity9

67Ca#ity967Catc.Ca#ity967Toot.ac.eCa#ity9

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Conditional Inde'endence

! 67Ca#ity:Catc.:Toot.ac.e9

67Catc.967Ca#ity:Toot.ac.eCatc.9u.@

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(aye&ian (elief *et&

! A com'act notation for re're&entingconditional inde'endence a&&um'tion&

and .ence a com'act $ay ofre're&enting a oint di&tri%ution! Synta=-

A directed acyclic gra'.: one node 'er#aria%le

,ac. node augmented $it. localconditional 'ro%a%ility ta%le&

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(aye&ian (elief *et&

! *ode& $it. no incoming arc& 7rootnode&9 &im'ly .a#e 'rior& a&&ociated

$it. t.em! *ode& $it. incoming arc& .a#e ta%le&

enumerating t.e

67*odeConunction of 6arent&9 >.ere 'arent mean& t.e node at t.e

ot.er end of t.e incoming arc

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Alarm ,=am'le

! aria%le&- (urglar: MaryCall&:Jo.nCall&: ,art.?ua)e: Alarm

! *et$or) to'ology ca'ture& t.edomain cau&ality 7conditionalinde'endence a&&um'tion&9

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Alarm ,=am'le

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(aye&ian (elief *et&-

Semantic&! T.e full oint di&tri%ution for t.e *

#aria%le& in a (elief *et can %e reco#eredfrom t.e information in t.e ta%le&

P(X1,...XN) = P(Xi| Parents(Xi))i=1

N

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(elief *et Semantic&

Alarm ,=am'le! >.at are t.e c.ance& of Jo.n call&:

Mary call&: alarm i& going off: no

%urglary: no eart.?ua)e@

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Alarm ,=am'le

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Alarm ,=am'le

! 67J

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,#ent&

! >.at a%out nonFatomic e#ent&@! "emem%er to 'artition Any e#ent can

%e defined a& a com%ination of ot.ermore $ellF&'ecified e#ent& 67A9 67A

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,#ent&

! 67M

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,#ent&

! o$ a%out 67MAlarm9@ Tric) ?ue&tion t.atH& &omet.ing $e )no$

! o$ a%out 67M,art.?ua)e9@ *ot directly in t.e net$or)

re$rite a&

67M

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Sim'ler ,=am'le&

! LetH& &ay $e .a#e t$o #aria%le& A and (: and$e )no$ ( influence& A

! >.atH& 67A

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Sim'le ,=am'le

! *o$ I tell you t.at ( .a& .a''ened

! >.atH& you %elief in A@B

A

P(B)

P(A|B)

P(A|~B)

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Sim'le ,=am'le

! Su''o&e in&tead I &ay A .a& .a''ened

! >.atH& you %elief in (@B

A

P(B)

P(A|B)P(A|~B)

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Sim'le ,=am'le

! 67(A967(

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C.ain "ule (a&i&

67(:,:A:J:M9

67M(:,:A:J967(:,:A:J9

67J(:,:A967(:,:A9

67A(:,967(:,9

67(,967,9

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C.ain "ule (a&i&

! 67(:,:A:J:M9! 67M(:,:A:J967J(:,:A967A(:,967(,967,9

! 67MA9 67JA9 67A(:,967(967,9

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Alarm ,=am'le

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+etail&

! >.ere do t.e gra'.& come from@ Initially: t.e intuition& of domain e='ert&

! >.ere do t.e num%er& come from@ o'efully: from .ard data Sometime& from e='ert& intuition&

! o$ can $e com'ute t.ing& efficiently@ ,=actly %y not redoing t.ing& unnece&&arily (y a''ro=imating t.ing&

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(rea)

! "eading& for 'ro%a%ility 1B- All

14-! 432F438: 500: Sec 144

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*oi&yF;r! ,#en $it. t.e reduction in t.e num%er

of 'ro%a%ilitie& needed itH& .ard toaccumulate all t.e num%er& you need

! ,&'ecially true $.en &ome e#idence#aria%le& are &.ared among manycau&e&

! T.e *oi&yF;r .ac) i& a u&eful &.ortFcut! 67AC1

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*oi&yF;r

Cold Flu Malaria

Fever

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*oi&y ;r

! 67e#erCold9

! 67e#erMalaria9

! 67e#erlu9

! 67De#erCold9

! 67De#erMalaria9

! 67De#erlu9

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*oi&y ;r

! >.at doe& it mean for t.e tooccur@

! It mean& t.e cau&e $a& true and t.e&ym'tom didnHt .a''en

! >.atH& t.e 'ro%a%ility of t.at@

67De#erCau&e9! 67De#erlu9: etc

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*oi&y ;r

! If all t.ree cau&e& are true and you donHt.a#e a fe#er t.en all t.ree %loc)er&

are in effect! >.atH& t.e 'ro%a%ility of t.at@

67De#erflu:cold:malaria9

67De#erflu967De#ercold967De#ermalaria9! (ut 1 t.at 67e#ercau&e&9

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Com'uting $it. ((*&

! *ormal &cenario ou .a#e a %elief net con&i&ting of a %unc.

of #aria%le&! Some of $.ic. you )no$ to %e true 7e#idence9

! Some of $.ic. youHre a&)ing a%out 7?uery9

! Some you .a#enHt &'ecified 7.idden9

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,=am'le

! 6ro%a%ility t.at t.ereH& a %urglarygi#en t.at Jo.n and Mary are calling

! 67(J:M9 ( i& t.e ?uery #aria%le

J and M are e#idence #aria%le&

A and , are .idden #aria%le&

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,=am'le

! 6ro%a%ility t.at t.ereH& a %urglary gi#en t.at Jo.nand Mary are calling

! 67(J:M9 al'.a 67(:J:M9

al'.a E

67(:J:M:A:,9 G

67(:J:M:DA:,9G

67(:J:M:A:D,9G 67(:J:M: DA:D,9

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rom t.e *et$or)

e a AMPAJPEBAPEPBP )|()|(),|()()(

e a AMPAJPEBAPEPBP )|()|(),|()()(

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,='re&&ion Tree

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S'eedu'&

! +onHt recom'ute t.ing& +ynamic 'rogramming

! +onHt com'ute &ome t.ing& at all Ignore #aria%le& t.at canHt effect t.e

outcome

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,=am'le

! Jo.n call& gi#en%urglary

! 67J(9

e a m AMPaJPEBAPEPBP )|()|(),|()()(

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aria%le ,limination

! ,#ery #aria%le t.at i& not an ance&torof a ?uery #aria%le or an e#idence

#aria%le i& irrele#ant to t.e ?uery

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*e=t Time

! ini&. C.a'ter& 1B and 14