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tasks Inference ) e = E|i X( P rginal ma r osterio p compute : queries Simple ) se al f = ts tar S on, = hts Lig , empty =e Gaug | oGas N( P e.g., ) e = E,i X|j X( P) e = E|i X( P =) e = E|j X,i X( P: queries Conjunctive rmation; info y utilit include rks ow net decision : decisions Optimal ) idence ev action, | outcome ( Pr fo required inference robabilistic p next? seek to evidence which : rmation info of alue V critical? most re a values y robabilit p which : analysis y Sensitivit r? moto rter sta new a need I do why : Explanation 3 14.4–5 Chapter
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umeration en yb Inference actually without joint the from riables va out sum to ya w intelligent Slightly resentation rep explicit its constructing rk: ow net ry burgla the on query Simple E B J A M ) mj, | B( P ) mj, ( /P ) mj, , B( P = ) mj, , B( Pα = ) mj, a, e, , B( Pa Σe Σα = entries: CPT of duct ro p using entries joint full Rewrite ) mj, | B( P ) a| m( P) a| j( P)e, B| a( P)e( P) B( Pa Σe Σα = ) a| m( P) a| j( P)e, B| a( Pa Σ)e( Pe Σ) B( Pα = d( O space, ) n( O enumeration: depth-first Recursive n time ) 4 14.4–5 Chapter
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algorithm umeration En X over distribution a returns ) bn , e, X( tion-Ask Enumera function riable va query the , X: inputs E riables va r fo values observed , e Y ∪ E ∪ } X{ riables va with rk ow net esian y Ba a , bn y empt initially , X over distribution a ←) X( Q do X of i x value h eac for Xr fo i x value with e extend ) e ], bn [ ars V( te-All Enumera ←)i x( Q )) X( Q( Normalize return er numb real a returns ) e, vars ( te-All Enumera function 1.0 return then ) vars ( Empty? if ) vars ( First ←Y e in y value has Y if ) e ), vars ( Rest ( te-All Enumera × )) Y( a P| y( P return then return else P )y e ), vars ( Rest ( te-All Enumera × )) Y( a P| y( P y y = Y with extended e is y e where 5 14.4–5 Chapter
9 9-PPI a jo anjen ypea oj (w)d(D)d saindwo a uoneindwo paleadal :qualpiyaul si uonejawnu 10 OL IO OL oLl四d (ol四d (oLl四d (ou叫d SO 06 SO 06 DL9d Dd Ld ol9d Q 90 6 SO S6 (aLqloud (aL'ql)d a'qp→d ⊕ (a'ql)d 866 Z00 (aud ⊕ (a)d I00 Qd O a017 uorlenjeAH
tree aluation Ev P(j|a) .90 P(m|a) .01 .70 P(m| a) .05 P(j|a) P(j| a) .90 P(m|a) .01 .70 P(m| a) .05 P(j| a) P(b) .001 P(e) .002 P( e) .998 P(a|b,e) .06 .95 P( a|b, e) .05 P( a|b,e) .94 P(a|b, e) computation eated rep inefficient: is Enumeration e of value each r fo ) a| m( P) a| j( P computes e.g., 6 14.4–5 Chapter
L 9PI duD (q)NYr×(q)ao= (d ano wns)(q)nrYaf(g)do= (v ino wns)('q)wrYf(a)(g)do= (D)nf()rf(q')f()d(g)o= (o)wf(o)ff(a‘ap)d"☒(o)d☒(a)do= (o)mf(ol)d(3aD)d”☒e)dK(a)do= 8 (olu)d(ol)d(3‘alp)d☒(a)d(a)do= (u‘Cla)d uoleindwooa pIone o1 (siopej)sa]nsal azelpawaul Sulons 1aj-01-1y3u suonewwns ino Kue :uoneulwlja a]qeue uorzeurw!e alqerieA Aq aouolojul
elimination ariable v yb Inference right-to-left, summations out rry ca elimination: riable aV recomputation avoid to ) rs facto ( results intermediate ring sto ) mj, | B( P ) B( Pα = } {z | B )e( Pe Σ } {z | E )e, B| a( Pa Σ } {z | A ) a| j( P } {z | J ) a| m( P } {z | M ) a( Mf) a| j( P)e, B| a( Pa Σ)e( Pe Σ) B( Pα = ) a( Mf) a( J f)e, B| a( Pa Σ)e( Pe Σ) B( Pα = ) a( Mf) a( J f)e b, a, ( Afa Σ)e( Pe Σ) B( Pα = ) A out (sum )e b, ( MJ¯Af)e( Pe Σ) B( Pα = ) E out (sum )b( M ¯AJ ¯Ef) B( Pα = )b( M ¯AJ ¯Ef ×)b( Bfα = 7 14.4–5 Chapter
8 g-'PI duO (bqo)f=(bq)4×(9D)f:83 (Iz…Tzf.f..Ix)f= (1zTzf.f)对×(fTf5x…Tx) :f pue If soej jo npoud asIMqu!od X uo puadap aou op ff uiwnsse f×f×…×y=y×…×+y〖f×…×y=y×…×r区 sopey Bululewal jo onpod asiMquiod ul saouewqns dn ppe uolewwns ay]apisino sionpej quensuoo Kue anow :suonej jo ionpoud e wouy ajqeuen e ino gulwwns suorgelodo orseg :uoreuru!o olqeLie
erations op Basic elimination: ariable V rs: facto of duct ro p a from riable va a out Summing summation the outside rs facto constant any move rs facto remaining of duct ro p wise oint p in submatrices up add ¯ Xf ×i f ×· · · ×1 f =k f ×· · · × +1 i f x Σi f ×· · · ×1 f =k f ×· · · ×1 fx Σ X on end dep not do i f, . . . , 1 f assuming : 2 f and 1 f rs facto of duct ro p wise oint P )l z, . . . , 1 z, k y, . . . , 1 y(2 f ×)k y, . . . , 1 y, j x, . . . , 1 x(1 f )l z, . . . , 1 z, k y, . . . , 1 y, j x, . . . , 1 x( f = )c b, a, ( f =)c b, (2 f ×)b a, (1 f E.g., 8 14.4–5 Chapter
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algorithm elimination ariable V X over distribution a returns ) bn , e, X( tion-Ask Elimina function riable va query the , X: inputs event an as ecified sp evidence , e )n X, . . . , 1 X( P distribution joint ecifying sp rk ow net elief b a , bn ]) bn [ ars V( Reverse ← vars ;] [ ← factors do vars in var h eac for ] factors |) e, var ( ctor a Make-F [ ← factors ) factors , var ( Sum-Out ← factors then riable va hidden a is var if )) factors ( oduct Pointwise-Pr ( Normalize return 9 14.4–5 Chapter
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ariables v t an Irrelev ) ue tr =y ar l g ur B| sl al ohnC J( P query the Consider E B J A M )b( Pα =)b| J( P Xe )e( P Xa ) a| J( P)e b, | a( P Xm ) a| m( P query the to t an irrelev is M1; identically is m over Sum ) E∪} X{(s Ancestor ∈ Y unless irrelevant is Y 1: Thm and , }y ar l g ur B{ = E, sl al ohnC J = X Here, }e uak thq ar E m, ar Al { =) E∪} X{(s Ancestor irrelevant is sl al Cy ar M so KBs) clause rn Ho in query the from chaining rd a backw to this re (Compa 10 14.4–5 Chapter