BLOG Probabilistic Mo dels with Unkno wn Ob jects Brian Milc Computer Science Division Univ ersit of California at Berk eley USA milc hcs PDF document - DocSlides

BLOG Probabilistic Mo dels with Unkno wn Ob jects Brian Milc Computer Science Division Univ ersit of California at Berk eley USA milc hcs PDF document - DocSlides

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BLOG: Probabilistic Mo dels with Unkno wn Ob jects Brian Milc Computer Science Division Univ ersit of California at Berk eley USA milc h@cs.b erk eley .edu ttp://www.cs.b erk eley .edu/ milc Bhask ara Marthi Computer Science Division Univ ersit of California at Berk eley USA bhask ara@cs.b erk eley .edu ttp://www.cs.b erk eley .edu/ bhask ara Stuart Russell Computer Science Division Univ ersit of California at Berk eley USA russell@cs.b erk eley .edu ttp://www.cs.b erk eley .edu/ russell Da vid Son tag Departmen of Electrical Engineering and Computer Science Massac usetts Institute of ec hnology USA dson tag@csail.mit.edu ttp://p eople.csail.mit.edu/dson tag Daniel L. Ong Computer Science Division Univ ersit of California at Berk eley USA dlong@o cf.b erk eley .edu ttp://www.o cf.b erk eley .edu/ dlong Andrey Kolob Computer Science Division Univ ersit of California at Berk eley USA ara aone@ram bler.ru
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BLOG: Pr ob abilistic Mo dels with Unknown Obje cts 1.1 In tro duction Human eings and AI systems ust con ert sensory input in to some understanding of what is going on in the orld around them. That is, they ust mak inferences ab out the ob jects and ev en ts that underlie their observ ations. No pre-sp eciˇed list of ob jects is giv en; the agen ust infer the existence of ob jects that ere not kno wn initially to exist. In man AI systems, this problem of unkno wn ob jects is engineered or resolv ed in prepro cessing step. Ho ev er, there are imp ortan applications where the problem is una oidable. Population estimation for example, in olv es coun ting opulation sampling from it randomly and measuring ho often the same ob ject is resampled; this ould oin tless if the set of ob jects ere kno wn in adv ance. or linkage task undertak en an industry of more than 300 companies, in olv es matc hing en tries across ultiple databases. These companies exist ecause of uncertain ab out the mapping from observ ations to underlying ob jects. Finally multi-tar get tr acking systems erform data asso ciation connecting, sa radar blips to yp othesized aircraft. Probabilit mo dels for suc tasks are not new: Ba esian mo dels for data asso- ciation ha een used since the 1960s [Sittler, 1964]. The mo dels are written in English and mathematical notation and con erted hand in to sp ecial-purp ose co de. This can result in in—exible mo dels of limited expressiv eness|for example, trac king systems assume indep enden tra jectories with linear dynamics, and record link age systems assume naiv Ba es mo del for ˇelds in records. It seems natural, therefore, to seek formal language in whic to express probabilit mo dels that allo for unkno wn ob jects. Recen ac hiev emen ts in the ˇeld of probabilistic graphical mo dels [P earl, 1988] illustrate the eneˇts that can exp ected from adopting formal language: general-purp ose inference algorithms, more sophisticated mo dels, and tec hniques for automated mo del selection (structure learning). Ho ev er, graphical mo dels only describ ˇxed sets of random ariables with ˇxed dep endencies among them; they ecome wkw ard in scenarios with unkno wn ob jects. There has also een signiˇcan ork on ˇrst-or der pr ob abilistic languages (F OPLs), whic explicitly represen ob jects and the relations et een them. review some of this ork in Section 1.7. Ho ev er, most OPLs mak the assumptions of unique names requiring that the sym ols or terms of the language all refer to distinct ob jects, and domain closur requiring that no ob jects exist esides the ones referred to terms in the language. These assumptions are inappropriate for problems suc as ulti-target trac king, where ma an to reason ab out ob jects that are observ ed ultiple times or that are not observ ed at all. Those OPLs that do supp ort unkno wn ob jects often do so in limited and ad ho ys. In this hapter, describ Ba esian logic Blog [Milc et al., 2005a], new language that compactly and in tuitiv ely deˇnes probabilit distributions er outcomes with arying sets of ob jects.
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1.2 Examples egin in Section 1.2 with three example problems, eac of whic in olv es ossible orlds with arying ob ject sets and iden tit uncertain sho Blog mo dels for these problems and giv initial, informal descriptions of the probabilit distributions that they deˇne. Section 1.3 observ es that the ossible orlds in these scenarios are naturally view ed as mo del structures of ˇrst-or der lo gic It then deˇnes precisely the set of ossible orlds corresp onding to Blog mo del. The ey idea is generativ pro cess that constructs orld adding ob jects whose existence and prop erties dep end on those of ob jects already created. In suc pro cess, the existence of ob jects ma go erned man random ariables, not just single opulation size ariable. Section 1.4 discusses exactly ho Blog mo del sp eciˇes probabilit distribution er ossible orlds. Section 1.5 solv es previously unnoticed \probabilistic Sk olemization" problem: ho to sp ecify evidence ab out ob jects|suc as radar blips|that one didn't kno existed. Finally Section 1.6 brie—y discusses inference in un ounded outcome spaces, stating sampling algorithm and completeness theorem for large class of Blog mo dels and giving exp erimen tal results on one particular mo del. 1.2 Examples In this section examine three ypical scenarios with unkno wn ob jects|simpliˇed ersions of the opulation estimation, record link age, and ultitarget trac king problems men tioned ab e. In eac case, pro vide short Blog mo del that, when com bined with suitable inference engine, constitutes orking solution for the problem in question. Example 1.1 An urn con tains an unkno wn um er of balls|sa um er hosen from oisson distribution. Balls are equally lik ely to blue or green. dra some balls from the urn, observing the color of eac and replacing it. cannot tell iden tically colored balls apart; furthermore, observ ed colors are wrong with probabilit 0.2. Ho man balls are in the urn? as the same ball dra wn wice? The Blog mo del for this problem, sho wn in Figure 1.1, describ es sto hastic pro cess for generating orlds. The ˇrst lines in tro duce the yp es of ob jects in these orlds|colors, balls, and dra ws|and the functions that can applied to these ob jects. or eac function, the mo del sp eciˇes typ signatur in syn tax similar to that of or Ja a. or instance, line sp eciˇes that rueColo is random function that tak es single argumen of yp Ball and returns alue of yp Colo Lines 5{7 sp ecify what ob jects ma exist in eac orld. In ev ery orld, there are exactly distinct colors, blue and green, and there are exactly four dra ws. These are the guar ante ob jects. On the other hand, di˛eren orlds ha di˛eren um ers of balls, so the um er of balls that exist is hosen from prior|a oisson with mean 6. Eac ball is then giv en color, as sp eciˇed on line 8. Prop erties of the four dra ws are ˇlled in ho osing ball (line 9) and an observ ed color for that ball (lines
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BLOG: Pr ob abilistic Mo dels with Unknown Obje cts type Color; type Ball; type Draw; random Color TrueColor(Ball); random Ball BallDrawn(Draw); random Color ObsColor(Draw); guaranteed Color Blue, Green; guaranteed Draw Draw1, Draw2, Draw3, Draw4; #Ball Poisson[6](); TrueColor(b) TabularCPD[[0.5, 0.5]](); BallDrawn(d) Uniform( Ball ); 10 ObsColor(d) 11 if (BallDrawn(d) != null) then 12 TabularCPD[[0.8, 0.2], [0.2, 0.8]](TrueColor(BallDrawn(d))); Figure 1.1 Blog mo del for balls in an urn (Example 1.1) with four dra ws. 10{12). The probabilit of the generated orld is the pro duct of the probabilities of all the hoices made. Example 1.2 ha collection of citations that refer to publications in certain ˇeld. What publications and researc hers exist, with what titles and names? Who wrote whic publication, and to whic publication do es eac citation refer? or simplicit just consider the title and author-name strings in these citations, whic are sub ject to errors of arious kinds, and assume only single-author publications. Figure 1.2 sho ws Blog mo del for this example, based on the mo del in [P asula et al., 2003]. The Blog mo del deˇnes the follo wing generativ pro cess. First, sample the total um er of researc hers from some distribution; then, for eac researc her sample the um er of publications that researc her. Sample the researc hers' names and publications' titles from appropriate prior distributions. Then, for eac citation, sample the publication cited ho osing uniformly at random from the set of publications. Finally generate the citation text with \noisy" formatting distribution that allo ws for errors and abbreviations in the title and author names. Example 1.3 An unkno wn um er of aircraft exist in some olume of airspace. An aircraft's state (p osition and elo cit y) at eac time step dep ends on its state at the previous time step. observ the area with radar: aircraft ma app ear as iden tical blips on radar screen. Eac blip giv es the appro ximate osition of the aircraft that generated it. Ho ev er, some blips ma false detections, and some aircraft ma not detected. What aircraft exist, and what are their tra jectories? Are there an aircraft that are not observ ed?
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1.2 Examples type Researcher; type Publication; type Citation; random String Name(Researcher); random String Title(Publication); random Publication PubCited(Citation); random String Text(Citation); origin Researcher Author(Publication); guaranteed Citation Cite1, Cite2, Cite3, Cite4; #Researcher NumResearchersPrior(); #Publication(Author r) NumPubsPrior(); 10 Name(r) NamePrior(); 11 Title(p) TitlePrior(); 12 PubCited(c) Uniform( Publication ); 13 Text(c) NoisyCitationGrammar(Title(PubCited(c)), 14 Name(Author(PubCited(c)))); Figure 1.2 Blog mo del for Example 1.2 with four observ ed citations. type Aircraft; type Blip; random R6Vector State(Aircraft, NaturalNum); random R3Vector ApparentPos(Blip); nonrandom NaturalNum Pred(NaturalNum) Predecessor; origin Aircraft Source(Blip); origin NaturalNum Time(Blip); #Aircraft NumAircraftPrior(); State(a, t) if then InitState() 10 else StateTransition(State(a, Pred(t))); 11 #Blip(Source a, Time t) DetectionCPD(State(a, t)); 12 #Blip(Time t) NumFalseAlarmsPrior(); 13 ApparentPos(b) 14 if (Source(b) null) then FalseAlarmDistrib() 15 else ObsCPD(State(Source(b), Time(b))); Figure 1.3 Blog mo del for Example 1.3.
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BLOG: Pr ob abilistic Mo dels with Unknown Obje cts The Blog mo del for this scenario (Figure 1.3) describ es the follo wing pro cess: ˇrst, sample the um er of aircraft in the area. Then, for eac time step (starting at 0), ho ose the state (p osition and elo cit y) of eac aircraft giv en its state at time 1. Also, for eac aircraft and time step ossibly generate radar blip with Source and Time Whether blip is generated or not dep ends on the state of the aircraft|th us the um er of ob jects in the orld dep ends on certain ob jects' attributes. Also, at eac step generate some false alarm blips with Time and Source null Finally sample the osition for eac blip giv en the true state of its source aircraft (or using default distribution for false-alarm blip). 1.3 Syn tax and Seman tics: ossible orlds 1.3.1 Outcomes as ˇrst-order mo del structures The ossible outcomes for Examples through are structures con taining man related ob jects, with the set of ob jects and the relations among them arying from outcome to outcome. will treat these outcomes formally as mo del structur es of ˇrst-or der lo gic mo del structure pro vides in terpretations for the sym ols of ˇrst-order language; eac sen tence of the ˇrst-order language can ev aluated to yield truth alue in eac mo del structure. In Example 1.1, the language has function sym ols suc as rueColo for the true color of ball BallDra wn for the ball dra wn on dra and Dra w1 for the ˇrst dra w. (Usually ˇrst-order languages are describ ed as ha ving predicate, function, and constan sym ols. or conciseness, view all sym ols as function sym ols; predicates are just functions that return Bo olean alue, and constan ts are just zero-ary functions.) eliminate meaningless random ariables, use typ logical languages. Eac Blog mo del uses language with particular set of yp es, suc as Ball and Dra Blog also has some built-in yp es that are ailable in all mo dels, namely Bo olean NaturalNum Integer String Real and ecto (for eac 2). Eac function sym ol has typ signatur ), where is the return yp of and are the argumen yp es. The yp Bo olean receiv es sp ecial syn tactic treatmen t: if the return yp of function is Bo olean then terms of the form constitute atomic form ulas, whic can com bined using logical op erators and placed inside quan tiˇers. The logical languages used in Blog are also fr function is not required to apply to all tuples of argumen ts, ev en if they are appropriately yp ed [Lam ert, 1998]. or instance, in Example 1.3, the function Source usually maps blips to aircraft, but it is not applicable if the blip is false detection. adopt the con en tion that when function is not applicable to some argumen ts, it returns the sp ecial alue null An function that receiv es null as an argumen also returns null and an atomic form ula that ev aluates to null is treated as false. The truth of an ˇrst-order sen tence is determined mo del structur for the
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1.3 Syntax and Semantics: Possible Worlds corresp onding language. mo del structure sp eciˇes the extension of eac yp and the interpr etation for eac function sym ol: Deˇnition 1.1 mo del structure of yp ed, free, ˇrst-order language consists of an extension for eac yp whic ma an arbitrary set, and an in terpretation for eac function sym ol If has return yp and argumen yp es then is function from to null (a) (b) (c) Figure 1.4 Three mo del structures for the language of Figure 1.1. Shaded circles represen balls that are blue; shaded squares represen dra ws where the dra wn ball app eared blue (unshaded means green). Arro ws represen the BallDra wn function from dra ws to balls. Three mo del structures for the language used in Figure 1.1 are sho wn in Fig- ure 1.4. Iden tit uncertain arises ecause BallDra wn Dra w1 migh equal to BallDra wn Dra w2 in one structure (suc as Figure 1.4(a)) but not another (suc as Figure 1.4(b)). The set of balls, Ball can also ary et een structures, as Figure 1.4 illustrates. The purp ose of Blog mo del is to deˇne probabilit distribution er suc structures. Because an sen tence can ev aluated as true or false in eac mo del structure, distribution er mo del structures implicitly deˇnes the probabilit that is true for eac sen tence in the logical language. 1.3.2 Outcomes with ˇxed ob ject sets egin our formal discussion of Blog seman tics considering the relativ ely simple case of mo dels with ˇxed sets of ob jects. Blog mo dels for ˇxed ob ject sets ha ˇv kinds of statemen ts. typ de clar ation suc as the statemen ts on line of Figure 1.3, in tro duces yp e. andom function de clar ation suc as line of Figure 1.3, sp eciˇes the yp signature for function sym ol whose alues will hosen randomly in the generativ pro cess. nonr andom function deˇnition
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BLOG: Pr ob abilistic Mo dels with Unknown Obje cts suc as the one on line of Figure 1.3, in tro duces function whose in terpretation is ˇxed in all ossible orlds. In our implemen tation, the in terpretation is giv en Ja class Predecessor in this example). guar ante obje ct statement suc as line in Figure 1.1, in tro duces and names some distinct ob jects that exist in all ossible orlds. or the built-in yp es, the ob vious sets of guaran teed ob jects and constan sym ols are predeˇned. The set of guaran teed ob jects of yp in Blog mo del is denoted ). Finally for eac random function sym ol, Blog mo del includes dep endency statement sp ecifying ho alues are hosen for that function. ostp one further discussion of dep endency statemen ts to Section 1.4. The ˇrst four kinds of statemen ts listed ab deˇne particular yp ed ˇrst-order language for mo del The set of ossible worlds of denoted consists of those mo del structures of where the extension of eac yp is ), and all nonrandom function sym ols (including guaran teed constan ts) ha their giv en in terpretations. or eac random function and tuple of appropriately yp ed guar- an teed ob jects can deˇne random ariable (R V) ). or instance, in simpliˇed er- sion of Example 1.1 where the urn con tains kno wn set of balls Ball1 Ball8 and mak four dra ws, the random ariables are rueColo Ball1 rueColo Ball8 ], BallDra wn Dra w1 BallDra wn Dra w4 ], and ObsColo Dra w1 ObsColo Dra w4 The ossible orlds are in one-to-one corresp ondence with full instan tiations of these basic Vs. Th us, join distribution for the basic Vs deˇnes distribution er ossible orlds. 1.3.3 Unkno wn ob jects In general, Blog mo del deˇnes generativ pro cess in whic ob jects are added iterativ ely to orld. describ suc pro cesses, ˇrst in tro duce origin function de clar ations suc as lines 5{6 of Figure 1.3. Unlik other functions, origin functions suc as Source or Time ha their alues set when an ob ject is added. An origin function ust tak single argumen of some yp (namely Blip in the example); it is then called -origin function. Generativ steps that add ob jects to the orld are describ ed numb er state- ments suc as line 11 of Figure 1.3: #Blip(Source a, Time t) DetectionCPD(State(a, t)); This statemen sa ys that for eac aircraft and time step the pro cess adds some um er of blips, and eac of these added blips has the prop ert that Source and Time In general, the eginning of um er statemen has the form: 1. In [Milc et al., 2005a] used the term \generating function", but ha no adopted the term \origin function" ecause it seems clearer.
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1.3 Syntax and Semantics: Possible Worlds where is yp e, are -origin functions, and are logical ariables. (F or yp es that are generated ab initio with no origin functions, the empt paren theses are omitted, as in Figure 1.1.) The inclusion of um er statemen means that for eac appropriately yp ed tuple of ob jects the generativ pro cess adds some random um er (p ossibly zero) of ob jects of yp suc that for Note that the yp es of the generating ob jects are the return yp es of Ob ject generation can ev en recursiv e: ob jects can generate other ob jects of the same yp e. or instance, consider mo del of sexual repro duction in whic ev ery male{female pair of individuals pro duces some um er of o˛spring. could represen this with the um er statemen t: #Individual(Mother m, Father f) if Female(m) !Female(f) then NumOffspringPrior(); can also view um er statemen ts more declarativ ely: Deˇnition 1.2 Let mo del structure of and consider um er statemen for yp with origin functions An ob ject satisˇes this um er statemen applied to in if for and null for all other -origin functions Note that if um er statemen for yp omits one of the -origin functions, then this function tak es on the alue null for all ob jects satisfying that um er statemen t. or instance, Source is null for ob jects satisfying the false-detection um er statemen on line 12 of Figure 1.3: #Blip(Time t) NumFalseAlarmsPrior(); Also, Blog mo del cannot con tain um er statemen ts with the same set of origin functions. This ensures that, in an giv en mo del structure, eac ob ject has exactly one generation history whic can found tracing bac the origin functions on The set of ossible orlds is the set of mo del structures that can constructed 's generativ pro cess. complete the picture, ust explain not only how many ob jects are added on eac step, but also what these ob jects are. It turns out to con enien to deˇne the generated ob jects as follo ws: when um er statemen with yp and origin functions is applied to generating ob jects the generated ob jects are tuples where is the um er of ob jects generated. Th us in Example 1.3, the aircraft are pairs Aircraft 1), Aircraft 2), etc., and the blips generated aircraft are nested tuples suc as Blip Source Aircraft 2)) Time 8) 1). The tuple enco des the ob ject's generation history; of course, it is purely in ternal to the seman tics and remains in visible to the user.
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10 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts Deˇnition 1.3 The universe of yp in Blog mo del denoted ), consists of the guaran teed ob jects of yp as ell as all nested tuples of yp that can generated from the guaran teed ob jects through ˇnitely man recursiv applications of um er statemen ts. As the follo wing deˇnition stipulates, in eac ossible orld the extension of is some subset of Deˇnition 1.4 or Blog mo del the set of ossible orlds is the set of mo del structures of suc that: 1. for eac yp 2. nonrandom functions ha the sp eciˇed in terpretations; 3. for eac um er statemen in with yp and origin functions and eac appropriately yp ed tuple of generating ob jects in the set of ob jects in that satisfy this um er statemen applied to these generating ob jects is for some natural um er 4. for ev ery yp eac elemen of satisˇes some um er statemen applied to some ob jects in Note that part of this deˇnition, the um er of ob jects generated an giv en application of um er statemen in orld is ˇnite um er Ho ev er, orld can still con tain inˇnitely man non-guaran teed ob jects if some um er statemen ts are applied recursiv ely: then the orld ma con tain tuples that are nested to depths with no upp er ound. Inˇnitely man ob jects can also result if um er statemen ts are triggered for ev ery natural um er, lik the statemen ts that generate radar blips in Example 1.3. With ˇxed set of ob jects, it as easy to deˇne set of basic Vs suc that full instan tiation of the basic Vs uniquely iden tiˇed ossible orld. ac hiev the same e˛ect with unkno wn ob jects, need kinds of basic Vs: Deˇnition 1.5 or Blog mo del the set of asic andom variables consists of: for eac random function with yp signature and eac tuple of ob jects ), function applic ation that is equal to if all exist in and null otherwise; for eac um er statemen with yp and origin functions that ha return yp es and eac tuple of ob jects numb er equal to the um er of ob jects that satisfy this um er statemen applied to in In tuitiv ely eac step in the generativ orld-construction pro cess determines the
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1.4 Syntax and Semantics: Pr ob abilities 11 alue of basic ariable. The crucial result ab out basic Vs is the follo wing: Pr op osition 1.6 or an Blog mo del and an complete instan tiation of there is at most one mo del structure in consisten with this instan tiation. Some instan tiations of do not corresp ond to an ossible orld: for ex- ample, an instan tiation for the urn-and-balls example where Ball 2, but rueColo Ball 7) is not null Instan tiations of that corresp ond to orld are called achievable Th us, to deˇne probabilit distribution er it suces to deˇne join distribution er the ac hiev able instan tiations of No that ha seen this tec hnical dev elopmen t, can sa more ab out the need to represen ob jects as tuples that enco de generation histories. Equat- ing ob jects with tuples migh seem unnecessarily complicated, but it ecomes ery helpful when deˇne Ba es net er the basic Vs (whic do in Section 1.4.2). or instance, in the aircraft trac king example, the paren of Appa rentP os Blip Source Aircraft 2)) Time 8) 1) is State Aircraft 2) It migh seem more elegan to assign um ers to ob jects as they are generated, so that the extension of eac yp in eac ossible orld ould simply preˇx of the natural um ers. Sp eciˇcally could um er the aircraft arbitrarily and then um er the radar blips lexicographically aircraft and time step. Then ould ha basic Vs suc as Appa rentP os 23 ], represen ting the apparen aircraft osition for blip 23. But blip 23 could generated an aircraft at an time step. In fact, the paren ts of Appa rentP os [23 ould ha to include all the Blip and State ariables in the mo del. So deˇning ob jects as tuples yields uc simpler Ba es net. 1.4 Syn tax and Seman tics: Probabilities 1.4.1 Dep endency statemen ts Dep endency and um er statemen ts sp ecify exactly ho the steps are carried out in our generativ pro cess. Consider the dep endency statemen for State a; from Figure 1.3: State(a, t) if then InitState() else StateTransition(State(a, Pred(t))); This statemen is applied for ev ery basic of the form State a; where Aircraft and If 0, the conditional distribution for State a; is giv en the elementary CPD InitState otherwise it is giv en the elemen tary CPD StateTransition whic tak es State Pred )) as an argumen t. These elemen tary CPDs deˇne distributions er ob jects of yp R6V ecto (the return yp of State ).
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12 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts In our implemen tation, elemen tary CPDs are Ja classes with metho getProb that returns the probabilit of particular alue giv en list of CPD argumen ts, and metho sampleVal that samples alue giv en the CPD argumen ts. dep endency statemen egins with function sym ol and tuple of logical ariables represen ting the argumen ts to this function. In um er statemen t, the ariables represen the generating ob jects. In either case, the rest of the statemen consists of sequence of clauses When the statemen is not abbreviated, the syn tax for the ˇrst clause is: if ond then elem-cp ar g1 ar gN The ond ortion is form ula of the ˇrst-order logical language (con taining no free ariables other than sp ecifying the condition under whic this clause should used to sample alue for basic V. More precisely if the ossible orld constructed so far is then the applicable clause is the ˇrst one whose condition is satisˇed in (assuming for the momen that is complete enough to determine the truth alues of the conditions). If no clause's condition is satisˇed, or if the basic refers to ob jects that do not exist in then the alue is set default to false for Bo olean functions, null for other functions, and zero for um er ariables. If the condition in clause is just true ", then the whole string if true then ma omitted. In the applicable clause, eac CPD argumen is ev aluated in The resulting alues are then passed to the elemen tary CPD. In the simplest case, the argumen ts are terms or form ulas of suc as State a; Pred )). An argumen can also set expr ession of the form where is yp e, is logical ariable, and is form ula. The alue of suc an expression is the set of ob jects suc that satisˇes with ound to If the form ula is just true it can omitted: this is the case on line of Figure 1.1, where just see the expression Ball Blog also includes other kinds of argumen ts to allo coun ting the um er of elemen ts in set, aggregating ultiset of alues, or passing in set of pairs o; where the 's are ob jects and the 's are non-uniform sampling eigh ts. require that the elemen tary CPDs ob ey rules related to non-guaran teed ob jects. First, if CPD is deˇning distribution er non-guaran teed ob jects e.g. the Uniform CPD on line of Figure 1.1), it should nev er assign ositiv probabilit to ob jects that do not exist in the partially completed orld ensure this, allo an elemen tary CPD to assign ositiv probabilit to non-guaran teed ob ject only if the ob ject as passed in as part of CPD argumen (in Figure 1.1, Ball is passed in). Second, an elemen tary CPD cannot \p eek" at the tuple represen tations of ob jects that are passed in: it ust in arian to erm utations of the non-guaran teed ob jects. 1.4.2 Declarativ seman tics So far ha explained Blog seman tics pro cedurally in terms of generativ pro cess. facilitate oth kno wledge engineering and the dev elopmen of learning
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1.4 Syntax and Semantics: Pr ob abilities 13 algorithms, ould lik to ha declarativ seman tics. The standard approac whic is used in most existing OPLs is to sa that Blog mo del deˇnes certain Ba esian net ork (BN) er the basic Vs. In this section discuss ho that approac needs to mo diˇed for Blog will write to denote an instan tiation of set of Vs ars ), and to denote the alue that assigns to If BN is ˇnite, then the probabilit it assigns to eac complete instan tiation is ars( a( ), where is the CPD for and a( is restricted to the paren ts of In an inˇnite BN, can write similar expression for eac ˇnite instan tiation that is closed under the paren relation (that is, ars( implies ars )). If the BN is acyclic and eac ariable has ˇnitely man ancestors, then these probabilit assignmen ts deˇne unique distribution [Kersting and De Raedt, 2001]. Figure 1.5 Ba es net for the Blog mo del in Figure 1.1. The ellipses and dashed arro ws indicate that there are inˇnitely man rueColo no des. The dicult is that in the BN corresp onding to Blog mo del, ariables often ha inˇnite paren sets. or instance, the BN for Example 1.1 (sho wn partially in Figure 1.5) has an inˇnite um er of basic Vs of the form rueColo ]: if it had only ˇnite um er of these Vs, it could not represen outcomes with more than balls. urthermore, eac of these rueColo Vs is paren of eac ObsColo V, since if BallDra wn happ ens to then the observ ed color on dra dep ends directly on the color of ball So the ObsColo no des ha inˇnitely man paren ts. In suc mo del, assigning probabilities to ˇnite instan tiations that are closed under the paren relation do es not deˇne unique distribution: in particular, it tells us nothing ab out the ObsColo ariables. required instan tiations to closed under the paren relation so that the factors a( ould ell-deˇned. But ma not need the alues of al of 's paren ts in order to determine the conditional distribution for or instance, kno wing BallDra wn Ball 13) and rueColo Ball 13) Blue is suf- ˇcien to determine the distribution for ObsColo the colors of all the other balls are irrelev an in this con text. can read o this con text-sp eciˇc indep endence from the dep endency statemen for ObsColo in Figure 1.1 noting that the in-
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14 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts stan tiation BallDra wn Ball 13), rueColo Ball 13) Blue determines the alue of the sole CPD argumen rueColo BallDra wn )). sa this instan tiation supp orts the ariable ObsColo (see [Milc et al., 2005b]). Deˇnition 1.7 An instan tiation supp orts basic of the form or if all ossible orlds consisten with agree on (1) whether all the ob jects exist, and, if so, on (2) the applicable clause in the dep en- dency or um er statemen for and the alues for the CPD argumen ts in that clause. Note that some Vs, suc as Ball in Example 1.1, are supp orted the empt instan tiation. can no generalize the notion of eing closed under the paren relation. Deˇnition 1.8 ˇnite instan tiation is self-supp orting if its instan tiated ariables can um- ered suc that for eac the restriction of to supp orts This deˇnition lets us giv seman tics to Blog mo dels in that is meaningful ev en when the corresp onding BNs con tain inˇnite paren sets. will write for the probabilit that 's dep endency or um er statemen assigns to the alue giv en an instan tiation that supp orts Deˇnition 1.9 distribution er satisˇes Blog mo del if for ev ery ˇnite, self- supp orting instan tiation with ars ( =1 ;:::;X (1.1) where is the set of ossible orlds consisten with and is um ering of as in Deˇnition 1.8. Blog mo del is wel l-deˇne if there is exactly one probabilit distribution that satisˇes it. Recall that BN is ell-deˇned if it is acyclic and eac ariable has ˇnite set of ancestors. Another of sa ying this is that eac ariable can \reac hed" en umerating its ancestors in ˇnite, top ologically ordered list. The ell-deˇnedness criterion for Blog is similar, but deals with ˇnite, self-supp orting instan tiations rather than ˇnite, top ologically ordered lists of ariables. Because are dealing with instan tiations rather than ariables, need to mak sure that they co er all ossible orlds in addition to co ering all basic ariables.
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1.4 Syntax and Semantics: Pr ob abilities 15 The or em 1.10 Let Blog mo del. Supp ose that is at most coun tably inˇnite, and for eac and there is self-supp orting instan tiation that agrees with and includes Then is ell-deˇned. Pr of pro vide only sk etc of the pro of here, deferring the full ersion to more tec hnical pap er. First, since is at most coun tably inˇnite, can imp ose an arbitrary um ering (a bijection with some preˇx of the natural um ers) on This um ering is \global" in the sense that it do es not dep end on the instan tiation of the random ariables. No w, deˇne sequence of auxiliary random ariables jg on as follo ws. Let where is the ˇrst basic in the global ordering that is supp orted the empt instan tiation. or 1, let the instan tiation )). Then let where is the ˇrst basic in the global ordering that is supp orted ), but has not already een used to deˇne for an The imp ortan prop ert of the sequence is that an instan tiation of determines the CPD for In other ords, if deˇne our mo del in terms of get standard BN in whic eac ariable has ˇnitely man ancestors. Ho ev er, ust sho that this sequence is ell-deˇned. Sp eciˇcally ust sho that for ev ery and ev ery there exists basic that is supp orted and has not already een used to deˇne for some This can sho wn using the premise that for ev ery there is self-supp orting instan tiation consisten with that con tains can use standard results from probabilit theory to sho that there is unique probabilit distribution er full instan tiations of suc that eac has the sp eciˇed conditional distribution giv en all its predecessors. It remains to sho that this distribution er instan tiations corresp onds to unique distribution on First, ust sho that eac full instan tiation of corresp onds to at most one ossible orld: this follo ws from Prop osition 1.6, plus the fact that full instan tiation of determines all the basic Vs. Second, can sho that the probabilit distribution ha deˇned er is concen trated on instan tiations that actually corresp ond to ossible orlds not instan tiations that giv Vs alues of the wrong yp e, or giv Vs non-n ull alues in con texts where they ust ull. Finally need to hec that this unique distribution on indeed satisˇes or ˇnite, self-supp orting instan tiations that corresp ond to the auxiliary instan- tiations used in deˇning the constrain is satisˇed construction. All other ˇnite, self-supp orting instan tiations can expressed as disjunctions of those \core" instan tiations. rom these observ ations, it is ossible to sho that (1.1) is satisˇed for all ˇnite, self-supp orting instan tiations. 2. This is satisˇed if the Real and ecto yp es are not argumen ts to random functions or return yp es of gorigin functions.
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16 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts hec that the criterion of Theorem 1.10 holds for particular example, need to consider eac basic V. In Example 1.1, the um er for balls is sup- orted the empt instan tiation, so in ev ery orld it is part of self-supp orting in- stan tiation of size one. Eac rueColo dep ends only on whether its argumen exists, so these ariables participate in self-supp orting instan tiations of size o. Similarly eac BallDra wn ariable dep ends only on what balls exist. sample an ObsColo ariable, need to kno BallDra wn and rueColo BallDra wn ], so these ariables are in self-supp orting instan tiations of size four. Similar argu- men ts can made for Examples 1.2 and 1.3. Of course, ould lik to ha an algorithm for hec king whether Blog mo del is ell-deˇned; the criteria giv en in Theorem 1.12 in Section 1.6.2 are ˇrst step in this direction. 1.5 Evidence and Queries Because ell-deˇned Blog mo del deˇnes distribution er mo del structures, can use arbitrary sen tences of as evidence and queries. But sometimes suc sen tences are not enough. In Example 1.3, the user observ es radar blips, whic are not referred to an terms in the language. The user could assert evidence ab out the blips using existen tial quan tiˇers, but then ho could he mak query of the form, \Did this blip come from the same aircraft as that blip?" natural solution is to allo the user to extend the language when evidence arriv es, adding constan sym ols to refer to observ ed ob jects. In man cases, the user observ es some new ob jects, in tro duces some new sym ols, and assigns the sym ols to the ob jects in an uninformativ order. handle suc cases, Blog includes sp ecial macro. or instance, giv en radar blips at time 8, one can assert: Blip r: Time(r) Blip1, Blip2, Blip3, Blip4 This asserts that there are exactly radar blips at time 8, and in tro duces new constan ts Blip1 Blip4 in one-to-one corresp ondence with those blips. ormally the macro augmen ts the mo del with dep endency statemen ts for the new sym ols. The statemen ts implemen sampling without replacemen t; for our example, ha Blip1 Uniform( Blip (Time(r) 8) ); Blip2 Uniform( Blip (Time(r) 8) (Blip1 != r) ); and so on. Once the mo del has een extended this the user can mak assertions ab out the apparen ositions of Blip1 Blip2 etc., and then use these sym ols in queries. These new constan ts resem ble Sk olem constan ts, but conditioning on assertions ab out the new constan ts is not the same as conditioning on an existen tial sen tence. or example, supp ose ou go in to new wine shop, pic up ottle at random, and observ that it costs $40. This scenario is correctly mo deled in tro ducing new
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1.6 Infer enc 17 constan Bottle1 with Uniform CPD. Then observing that Bottle1 costs at least $40 suggests that this is fancy wine shop. On the other hand, the mere existenc of $40+ ottle in the shop do es not suggest this, ecause almost ev ery shop has some ottle at er $40. 1.6 Inference Because the set of basic Vs of Blog mo del can inˇnite, it is not ob vious that inference for ell-deˇned Blog mo dels is ev en decidable. Ho ev er, the generativ pro cess in tuition suggests rejection sampling algorithm. presen this algorithm not ecause it is particularly ecien t, but ecause it demonstrates the decidabilit of inference for large class of Blog mo dels (see Theorem 1.12 elo w) and illustrates sev eral issues that an Blog inference algorithm ust deal with. the end of this section, presen exp erimen tal results from somewhat more ecien lik eliho eigh ting algorithm. 1.6.1 Rejection sampling Supp ose are giv en partial instan tiation as evidence, and query ariable generate eac sample, our rejection sampling algorithm starts with an empt instan tiation Then it rep eats the follo wing steps: en umerate the basic Vs in ˇxed order un til reac the ˇrst that is supp orted but not already instan tiated in sample alue for according to 's dep endency statemen t; and augmen with the assignmen The pro cess con tin ues un til all the query and evidence ariables ha een sampled. If the sample is consisten with the evidence then the program incremen ts coun ter where is the sampled alue of Otherwise, it rejects this sample. After accepted samples, the estimate of is This algorithm requires subroutine that determines whether partial instan- tiation supp orts basic and if so, returns sample from 's conditional distribution. or basic of the form or ], the subroutine egins hec king the alues of the relev an um er ariables in to determine whether all of exist. If some of these um er ariables are not instan tiated, then do es not supp ort If some of do not exist, the subroutine returns the default alue for If they do all exist, the subroutine follo ws the seman tics for dep endency statemen ts discussed in Section 1.4.1. First, it iterates er the clauses in the dep endency (or um er) statemen un til it reac hes 3. Eac basic or can assigned \depth" whic is the maxim um of the depths of nested tuples and the magnitudes of in tegers among its argumen ts The um er of Vs at eac giv en depth is ˇnite. Th us, can en umerate ˇrst the Vs at depth 0, then those at depth 1, depth 2, etc.
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18 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts clause whose condition is either undetermined or determined to true giv en (if all the conditions are determined to false, then it returns the default alue for ). If the condition is undetermined, then do es not supp ort If it is determined to true, then the subroutine ev aluates eac of the CPD argumen ts in this clause. If determines the alues of all the argumen ts, then the subroutine samples alue for passing those alues to the sampleVal metho of this clause's elemen tary CPD. Otherwise, do es not supp ort ev aluate terms and quan tiˇer-free form ulas, use straigh tforw ard recursiv algorithm. The base case lo oks up the alue of particular function application in if this is not instan tiated, the algorithm returns \undetermined". ev aluate form ula, ev aluate its subform ulas in order from left to righ t. stop when hit an undetermined subform ula or when the alue of the whole form ula is determined. or example, to ev aluate ˇrst ev aluate If is undetermined, return \undetermined"; if is true, return true and if is false, go on to ev aluate It is more complicated to ev aluate set expressions suc as Blip r: Time(r) whic can used as CPD argumen ts. naiv algorithm for ev aluating this ex- pression ould ˇrst en umerate all the ob jects of yp Blip (whic ould require certain um er ariables to instan tiated), then select the blips that satisfy Time 8. But Figure 1.3 sp eciˇes that there ma exist some blips for eac air- craft and eac natural um er since there are inˇnitely man natural um ers, some orlds con tain inˇnitely man blips. ortunately the um er of blips with Time is necessarily ˇnite: in ev ery orld there are ˇnite um er of aircraft, and eac one generates ˇnite um er of blips at time 8. ha an algorithm that scans the form ula within set expression for origin function estrictions suc as Time 8, and uses them to oid en umerating inˇnite sets when ossible. These restrictions ma either equalit constrain ts, or inequalities that deˇne ounded set of natural um ers, suc as Time 12. similar metho is used for ev aluating quan tiˇed form ulas. 1.6.2 ermination criteria In order to generate eac sample, the algorithm ab rep eatedly instan tiates the ˇrst ariable that is supp orted but not et instan tiated, un til it instan tiates all the query and evidence ariables. When can sure that this will tak ˇnite amoun of time? The ˇrst this pro cess could fail to terminate is if it go es in to an inˇnite lo op while hec king whether particular ariable is supp orted. This happ ens if the program ends up en umerating an inˇnite set while ev aluating set expression or quan tiˇed form ula. can oid this ensuring that all suc 4. This left-to-righ ev aluation sc heme do es not alw ys detect that form ula is deter- mined: for instance, on it returns \undetermined" if is undetermined but is true|ev en though ust true in this case.
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1.6 Infer enc 19 expressions in the Blog mo del are ˇnite once origin function restrictions are tak en in to accoun t. The sample generator also fails to terminate if it nev er constructs an instan tiation that supp orts particular query or evidence ariable. see ho this can happ en, consider calling the subroutine describ ed ab to sample ariable If is not supp orted, the subroutine will realize this when it encoun ters ariable that is relev an but not instan tiated. No consider graph er basic ariables where dra an edge from to when the ev aluation pro cess for hits in this If ariable is nev er supp orted, then it ust part of cycle in this graph, or part of receding hain of ariables that is extended inˇnitely The graph constructed in this aries from sample to sample: for instance, sometimes the ev aluation pro cess for ObsColo will hit rueColo Ball 7) ], and sometimes it will hit rueColo [( Ball 13) ]. Ho ev er, can rule out cycles and inˇnite receding hains in all these graphs considering more abstract graph er function sym ols and yp es (along the same lines as the dep endency gr aph of [Koller and Pfe˛er, 1998, riedman et al., 1999]). Deˇnition 1.11 The symb ol gr aph for Blog mo del is directed graph whose no des are the yp es and random function sym ols of where the paren ts of yp or function sym ol are: the random function sym ols that ccur on the righ hand side of the dep endency statemen for or some um er statemen for the yp es of ariables that are quan tiˇed er in form ulas or set expressions on the righ hand side of suc statemen t; the yp es of the argumen ts for or the return yp es of origin functions for The sym ol graphs for our three examples are sho wn in Figure 1.6. If the sampling subroutine for basic hits basic then there ust an edge from 's function sym ol (or yp e, if is um er V) to 's function sym ol (or yp e) in the sym ol graph. This prop ert along with ideas from [Milc et al., 2005b], allo ws us to pro the follo wing: The or em 1.12 Supp ose is Blog mo del where: 1. uncoun table built-in yp es do not serv as function argumen ts or as the return yp es of origin functions; 2. eac quan tiˇed form ula and set expression ranges er ˇnite set once origin function restrictions are tak en in to accoun t; 3. the sym ol graph is acyclic. Then is ell-deˇned. Also, for an evidence instan tiation and query ariable the rejection sampling algorithm describ ed in Section 1.6.1 con erges to the osterior deˇned the mo del, taking ˇnite time er sampling step.
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20 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts (a) (b) (c) Figure 1.6 Sym ol graphs for (a) the urn-and-balls mo del in Figure 1.1; (b) the bibliographic mo del in Figure 1.2; (c) the aircraft trac king mo del in Figure 1.3. The criteria in Theorem 1.12 are ery conserv ativ e: in particular, when construct the sym ol graph, ignore all structure in the dep endency statemen ts and just hec for the ccurrence of function and yp sym ols. These criteria are satisˇed the mo dels in Figures 1.1 and 1.2. Ho ev er, the aircraft trac king mo del in Figure 1.3 do es not satisfy the criteria ecause its sym ol graph (Figure 1.6(c)) con tains self-lo op from State to State The criteria do not exploit the fact that State a; dep ends only on State a; Pred )), and the nonrandom function Pred is acyclic. riedman et al. [1999] ha already dealt with this issue in the con text of probabilistic relational mo dels; their algorithm can adapted to obtain stronger ersion of Theorem 1.12 that co ers the aircraft trac king mo del. 1.6.3 Exp erimen tal results Milc et al. [2005b] describ guided lik eliho eigh ting algorithm that uses bac kw ard haining from the query and evidence no des to oid sampling irrelev an ariables. This algorithm can also adapted to Blog mo dels. applied this algorithm for Example 1.1, asserting that 10 balls ere dra wn and all app eared blue, and querying the um er of balls in the urn. Figure 1.7(a) sho ws that when the prior for the um er of balls is uniform er the osterior puts more eigh on small um ers of balls; this mak es sense ecause the more balls there
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1.7 elate Work 21 are in the urn, the less lik ely it is that they are all blue. Figure 1.7(b), using oisson(6) prior, sho ws similar but less pronounced e˛ect. Note that in Figure 1.7, the osterior probabilities computed the lik eliho eigh ting algorithm are ery close to the exact alues (computed exhaustiv en umeration of ossible orlds with up to 170 balls). ere able to obtain this lev el of accuracy using runs of 20,000 samples with the uniform prior, and 100,000 samples using the oisson prior. On Lin ux orkstation with 3.2 GHz en tium pro cessor, the runs with the uniform prior to ok ab out 35 seconds (571 samples/second), and those with the oisson prior to ok ab out 170 seconds (588 samples/second). Suc results could not obtained using an algorithm that constructed single ˇxed BN, since the um er of oten tially relev an rueColo ariables is inˇnite in the oisson case. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 1 2 3 4 5 6 7 8 Probability Number of balls in urn 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 5 10 15 20 25 Probability Number of balls in urn (a) (b) Figure 1.7 Distribution for the um er of balls in the urn (Example 1.1). Dashed lines are the uniform prior (a) or oisson prior (b); solid lines are the exact osterior giv en that 10 balls ere dra wn and all app eared blue; and plus signs are osterior probabilities computed indep enden runs of 20,000 samples (a) or 100,000 samples (b). 1.7 Related ork Gaifman [1964] as the ˇrst to suggest deˇning probabilit distribution er ˇrst- order mo del structures. Halp ern [1990] deˇnes language in whic one can mak statemen ts ab out suc distributions: for instance, that the probabilit of the set of orlds that satisfy Flies Tw eet is 0.8. Pr ob abilistic lo gic pr gr amming [Ng and Subrahmanian, 1992] can seen as an application of this approac to Horn-clause kno wledge bases. Suc an approac only deˇnes onstr aints on distributions, rather than deˇning unique distribution. Most ˇrst-order probabilistic languages (F OPLs) that deˇne unique distributions
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22 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts ˇx the set of ob jects and the in terpretations of (non-Bo olean) function sym ols. Ex- amples include relational Ba esian net orks [Jaeger, 2001] and Mark logic mo d- els [Domingos and Ric hardson, 2004]. Prolog-based languages suc as probabilistic Horn ab duction [P ole, 1993], PRISM [Sato and Kamey a, 2001], and Ba esian logic programs [Kersting and De Raedt, 2001] ork with Herbr and mo dels where the ob- jects are in one-to-one corresp ondence with the ground terms of the language (a consequence of the unique names and domain closure assumptions). There are few OPLs that allo explicit efer enc unc ertainty i.e., uncertain ab out the in terpretations of function sym ols. Among these are languages that use indexed Vs rather than logical notation: BUGS [Gilks et al., 1994] and indexed probabilit diagrams (IPDs) [Mjolsness, 2004]. Reference uncertain can also represen ted in probabilistic relational mo dels (PRMs) [Koller and Pfe˛er, 1998], where \single-v alued complex slot" corresp onds to an uncertain unary function. PRMs are unfortunately restricted to unary functions (attributes) and binary predicates (relations). Probabilistic en tit y-relationship mo dels [Hec erman et al., 2004] lift this restriction, but represen reference uncertain using relations (suc as Dra wn d; )) and sp ecial utual exclusivit constrain ts, rather than with functions suc as BallDra wn ). Multi-en tit Ba esian net ork logic (MEBN) [Lask ey and da Costa, 2005] is similar to Blog in allo wing uncertain ab out the alues of functions with an um er of argumen ts. The need to handle unkno wn ob jects has een appreciated since the early da ys of OPL researc h: Charniak and Goldman's plan recognition net orks (PRNs) [Char- niak and Goldman, 1993] can con tain un ounded um ers of ob jects represen ting yp othesized plans. Ho ev er, external rules are used to decide what ob jects and ariables to include in PRN. While eac ossible PRN deˇnes distribution on its wn, Charniak and Goldman do not claim that the arious PRNs are all appro ximations to some single distribution er outcomes. Some more recen OPLs do deˇne single distribution er outcomes with arying ob jects. IPDs allo uncertain er the index range for an indexed family of Vs. PRMs and their extensions allo ariet of forms of uncertain ab out the um er (or existence) of ob jects satisfying certain relational constrain ts [Koller and Pfe˛er, 1998, Geto or et al., 2001] or elonging to eac yp [P asula et al., 2003]. Ho ev er, there is no uniˇed syn tax or seman tics for dealing with unkno wn ob jects in PRMs. MEBNs tak et another approac h: an MEBN mo del includes set of unique iden tiˇers, for eac of whic there is an \iden tit y" indicating whether the named ob ject exists. Our approac to unkno wn ob jects in Blog can seen as unifying the PRM and MEBN approac hes. Num er statemen ts neatly generalize the arious ys of handling unkno wn ob jects in PRMs: um er uncertain [Koller and Pfe˛er, 1998] corresp onds to um er statemen with single origin function; existence uncertain [Geto or et al., 2001] can mo deled with or more origin functions (and CPD whose supp ort is ); and domain uncertain [P asula et al., 2003] corresp onds to um er statemen with no origin functions. There is also corresp ondence et een Blog and MEBN logic: the tuple represen tations in
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1.8 Conclusions and utur Work 23 Blog mo del can though of as unique iden tiˇers in an MEBN mo del. The di˛erence is that Blog determines whic ob jects actually exist in orld using um er ariables rather than individual existence ariables. Finally it is informativ to compare Blog with the IBAL language [Pfe˛er, 2001], in whic program deˇnes distribution er outputs that can arbitrary nested data structures. An IBAL program could implemen Blog -lik generativ pro cess with the outputs view ed as logical mo del structures. But the declarativ seman tics of suc program ould less clear than the corresp onding Blog mo del. 1.8 Conclusions and uture ork Blog is represen tation language for probabilistic mo dels with unkno wn ob jects. It con tributes to the solution of ery general problem in AI: in telligen systems ust represen and reason ab out ob jects, but those ob jects ma not kno wn priori and ma not directly and uniquely iden tiˇed erceptual pro cesses. Our approac deˇnes generativ mo dels in whic ˇrst-order mo del structures are created adding ob jects and setting function alues; ev erything else follo ws naturally from this design decision. Muc ork remains to done on Blog The inference algorithms presen ted in this pap er are not practical for an but the smallest examples. or real-w orld problems, exp ect to emplo Mark hain Mon te Carlo (MCMC) tec hniques (see, e.g, Gilks et al. [1996]), sim ulating Mark hain er ossible orlds. More precisely these algorithms ust use partial descriptions of ossible orlds: in mo del with inˇnitely man Vs, orld cannot represen ted explicitly as full instan tiation. plan to implemen general Gibbs sampling algorithm for Blog mo dels, using some of the same tec hniques as the BUGS system [Gilks et al., 1994]. Ho ev er, for mo dels with unkno wn ob jects, exp ect to obtain faster con ergence with Metrop olis-Hastings algorithms [Metrop olis et al., 1953] using prop osal distributions that split and merge ob jects [Jain and Neal, 2004]. or no w, it app ears that these prop osal distributions will need to designed hand to prop ose reasonable splits and merges (e.g., merging publications with similar or iden tical titles), as as done in [P asula et al., 2003]. Ho ev er, ha implemen ted general Metrop olis-Hastings inference engine for Blog that main tains the state of the Mark hain and computes acceptance probabilities for an giv en prop osal distribution. In the future, plan to explore adaptiv MCMC tec hniques (see, e.g., [Haario et al., 2001] and references therein). Another imp ortan question is ho to design Blog mo dels that will lead to ac- curate inferences from real-w orld data. or the citation matc hing problem, asula et al. [2003] obtained state-of-the-art accuracy using reasonably simple prior distri- butions for publication titles and author names, estimated from BibT eX ˇles and U.S. Census data (these results are comp etitiv with the discriminativ approac of ellner et al. [2004]). It is not so clear ho to estimate the prior distributions for the um ers of ob jects of arious yp es, suc as researc hers and publications.
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24 BLOG: Pr ob abilistic Mo dels with Unknown Obje cts asula et al. [2003] simply used log-normal distribution, whic has ery large ariance. As an alternativ to deˇning suc prior distribution, one could use the nonparametric ersion of Blog prop osed Carb onetto et al. [2005], whic incorp orates Diric hlet pro cess mixture mo dels. Finally erhaps the most in teresting questions ab out Blog ha to do with learning. arameter estimation, ev en from partially observ ed data, is conceptually straigh tforw ard: the sampling-based inference algorithms describ ed ab can serv as the basis for Mon te Carlo exp ectation-maximization (EM) algorithms [W ei and anner, 1990]. But learning the structure of Blog mo dels is an exciting op en problem. In other statistical relational formalisms, tec hniques ha een prop osed for disco ering dep endencies that hold et een attributes of related ob jects [F riedman et al., 1999, op escul et al., 2003]. eliev that extensions of these tec hniques can applied to Blog The ultimate goal, ho ev er, is to dev elop algorithms that can yp othesize new attributes, new relations, and ev en new yp es of ob jects. Blog pro vides language in whic suc yp otheses can expressed.
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