Markov Logic Network. Matthew Richardson and Pedro Domingos. IE , Present by Hao Wu (haowu4)
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1 Markov Logc Network Matthew Rchardson and Pedro Domngos IE , Present by Hao Wu (haowu4)
2 Motvaton - Unfyng Logc and Probablty Logc and probablty are two most mport way of reasonng. Classc AI favors logc approaches, whch s mostly rule based. Theorem proofng. Cannot deal wth uncertanty, very lmted success. Modern AI approaches are domnated by more probablstc methods, whch handles the uncertanty and nose n real data. Deep Learnng, PGM and etc. Huge success So why we stll want to have Logc? (Why not learn everythng?)
3 Why logc s stll nterestng Logc, especally, Frst-order logc provde a expressve, compact and elegant way to express knowledge. It only take 30+ lne to wrte down the rule of Sudoku n Prolog (and the same code can also solve t). How many data do you need to learn everythng from scratch? We want a nce way to represent and solve our problems (effcently). Use expert knowledge to help the data drven system. Markov Logc s a way to connects Logc and Probablty. Logc handles complexty. Probablty handles uncertanty.
4 Background: Markov Network Potental functons defned over clques 1 P ( x) = Φc( x c ) Z c Smokng Cancer Asthma Cough
5 Frst Order Logc Constants, varables, functons, predcates E.g.: Anna, x, MotherOf(x), Frends(x, y) Lteral: Predcate or ts negaton Clause: Dsjuncton of lterals Groundng: Replace all varables by constants E.g.: Frends (Anna, Bob) World (model, nterpretaton): Assgnment of truth values to all ground predcates
6 Comparson FOL : x x, y Smokes ( x) Frends ( x, y) Cancer ( x) ( Smokes ( x) Smokes ( y) ) MRF: Frends(A,B) Smokes(A) Smokes(B) Cancer(A) Cancer(B)
7 Markov Logc Network A Markov Logc Network (MLN) s a set of pars (F, w) where F s a formula n frst-order logc w s a real number x x, y Smokes ( x) Frends ( x, y) Cancer ( x) ( Smokes ( x) Smokes ( y) ) * And we need a database that contans constants for groundng.
8 x Smokes ( x) x, y Frends ( x, y) Cancer ( x) ( Smokes ( x) Smokes ( y) ) + Two constants: Anna (A) and Bob (B) Frends(A,B) Frends(A,A) Smokes(A) Smokes(B) Frends(B,B) Cancer(A) Frends(B,A) Cancer(B)
9 Markov Logc Network: Defnton Each ground formula defnes a clque s the number of true groundng of formula s the state (truth value) of atoms n formula
10 Markov Logc Networks A template for ground Markov Random Feld. Can have type to reduce the number of predcate X constants..e. Human can only be frend wth another human. Expressvty: When set all weght to nfnte large, t becomes FOL. Every probablty dstrbuton over dscrete or fnte- precson numerc varables can be represented as a Markov logc network.
11 Inference (Same as nference on MRF*) *Sometme need a lttle twst for MCMC style nference MAP Inference: arg max y P( Condtonal Inference y x)
12 Learnng Learn from a database Can to learn both weghts (parameters) and FOL formula(structure): Learnng weghts. By optmze lkelhood. Learnng formula: (Inductve Logc Programmng) An ILP system wll derve a hypothessedlogc program whch entals all the postve and none of the negatve examples. Use exstng Inductve logc programmng system.
13 Learnng weght Optmze lkelhood. (Generatve approach) f ( w) = log P( X = x) = Z = x Generalzed too hard, do Pseudo-lkelhood nstead. Countng true groundngs of a frst order clause n a KB s #P complete Optmze condtonal lkelhood. (Dscrmnatve approach) f ( w) = log P( Y = y X = x) = wn ( y, x) log Z x Z x = exp( wn ( y', x) ) y' exp ( w n ( x' )) = log P( X = l w n ( x) log Z log PL ( x) x MB( x )) l l l
14 Applcaton - Entty resoluton (Ctaton DB) Author(bb,author) Ttle(bb,ttle) Venue(bb,venue) HasWord(author,word) HasWord(ttle,word) HasWord(venue,word) SameAuthor(author1,author2) SameTtle(ttle1,ttle2) SameVenue(venue1,venue2) SameBb(bb1,bb2)
15 Applcaton - Entty resoluton Ttle(b1,t1) Ttle(b2,t2) HasWord(t1,+w) HasWord(t2,+w) SameBb(b1,b2) Author(b1,a1) Author(b2,a2) SameBb(b1,b2) SameAuthor(a1,a2) Author(b1,a1) Author(b2,a2) SameAuthor(a1,a2) Samebb(b1,b2)
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