A survey of first-order probabilistic models

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A Survey of First-Order Probabilistic Models

Rodrigo de Salvo Braz ,Eyal Amir,and Dan Roth

Department of Computer Science

University of Illinois at Urbana-Champaign

Urbana,IL61801

Summary.There has been a long standing division in Arti?cial Intelligence between logical and probabilistic reasoning approaches.While probabilistic models can deal well with inherent uncertainty in many real-world domains,they operate on a mostly propositional level.Logic systems,on the other hand,can deal with much richer rep-resentations,especially?rst-order ones,but treat uncertainty only in limited ways. Therefore,an integration of these types of inference is highly desirable,and many ap-proaches have been proposed,especially from the1990s on.These solutions come from many di?erent sub?elds and vary greatly in language,features and(when available at all)inference algorithms.Therefore their relation to each other is not always clear,as well as their semantics.In this survey,we present the main aspects of the solutions proposed and group them according to language,semantics and inference algorithm.In doing so,we draw relations between them and discuss particularly important choices and tradeo?s.

For decades after the?eld of Arti?cial Intelligence(AI)was established,its most prevalent form of representation and inference was logical,or at least symbolic representations that were in a deeper sense equivalent to a fragment of logic. While highly expressive,this type of model lacked a sophisticated treatment of degrees of uncertainty,which permeates real-world domains,especially the ones usually associated with intelligence,such as language,perception and common sense reasoning.

In time,probabilistic models became an important part of the?eld,incor-porating probability theory into reasoning and learning AI models.Since the 1980s the?eld has seen a surge of successful solutions involving large amounts of data processed from a probabilistic point of view,applied especially to Natural Language Processing and Pattern Recognition.1

Currently at the Computer Science Division of University of California,Berkeley. 1Strictly speaking,this tendency has not been only probabilistic,including machine learning methods such as neural networks that did not claim to be modeling prob-abilities.However,a link to probabilities can usually be found and the methods are used in similar ways.

D.E.Holmes and L.C.Jain(Eds.):Innovations in Bayesian Networks,SCI156,pp.289–317,2008. https://www.wendangku.net/doc/5b4326586.html, c Springer-Verlag Berlin Heidelberg2008

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This success,however,came with a price.Typically,probabilistic models are less expressive and?exible than logical or symbolic https://www.wendangku.net/doc/5b4326586.html,ually,they involve propositional,rather than?rst-order representations.When required, more expressive,higher level representations are obtained by ad hoc manipula-tions of lower level,propositional systems.

Starting in the1970s but having greatly increased from the1990s on,a line of research sought to integrate those two important modes of reasoning.In this chapter we give a survey of this research,and try to show some general lines separating di?erent approaches.

We have roughly divided this research in di?erent stages.The1970s and1980s saw great interest in expert systems[1,2].As these systems were applied to real-world domains,coping with uncertainty became more desirable,giving rise to the certainty factors approach,which uses rules with attached numbers(representing degrees of certainty)that get propagated to conclusions during inference.

Certainty factors systems did not have clear semantics,and often produced surprising and nonintuitive results[3].The search for clearer semantics for rules with varying certainty gave rise,among other things,to approaches such as Bayesian Networks.These however were essentially propositional,and thus had much less expressivity than logic systems.

The search for clear semantics of probabilities in logic systems resulted in works such as Nilsson[4],Bacchus[5]and Halpern[6],which laid out the basic theoretic principles supporting probabilistic logic.These works,however,did not include e?cient inference algorithms.

Works aiming at e?cient inference algorithms for?rst-order probabilistic in-ference(FOPI)can be divided in two groups,which Pearl[3]calls extensional and intensional systems.In the?rst one,statements in the language are more procedural in nature,standing for licenses for propagating truth values that have been generalized from true or false to a gray scale of varying degrees of certainty.In the second group,statements place restrictions on a probability distribution on possible worlds.They do not directly correspond to computing operations,nor can they typically be taken into account without regard to other rules(that is,inference is not completely modular).E?cient algorithms have to be devised for these languages that preserve their semantics while doing better than considering the entire model at every step.

Among intensional models,there are further divisions regarding the type of algorithm proposed.One group proposes inference rules similar to the ones used in?rst-order logic inference(for example,modus ponens).A second one com-putes,in more or less e?cient manners,the possible derivations of a query given a model.A third one uses sampling to answer queries about a model.A fourth and more prevalent group constructs a(propositional)graphical model(Bayesian or Markov networks,for example)that answers queries,and uses general graphical model inference algorithms for solving them.Finally,a?fth one proposes lifted algorithms that directly operate on?rst-order representations in order to derive answers to queries.

We now present these stages in more detail.

12A Survey of First-Order Probabilistic Models291 12.1Expert Systems and Certainty Factors

Expert systems are based on rules meant to be applied to existing facts,produc-ing new facts as conclusions[1].Typically,the context is a deterministic one in which facts and rules are assumed to be certain.Uncertainties from real-world applications are dealt with during the modeling stage where necessary(and often heavy-handed)simpli?cations are performed.

Certainty factors were introduced for the purpose of allowing uncertain rules and facts,making for more direct and accurate modeling.A rule(A←B):c1, with c1∈[0,1],indicates that we can conclude A with a degree of certainty of c1×c2,if B is known to be true with a degree of certainty c2∈[0,1].Given a

collection of rules and facts,inference is performed by propagating certainties in this fashion.There are also combination rules for the cases when more than one rule provide certainty factors for the same literal.

A paradigmatic application of certainty factors is the system MYCIN[7],an expert system dedicated to diagnosing diseases based on observed symptoms. Clark&McCabe[8]describe how to use Prolog with predicates containing an extra argument representing its certainty and being propagated accordingly. Shapiro[9]describes a Prolog interpreter that does the same but in a way im-plicit in the interpreter and language,rather than as an extra argument.

One can see that certainty factors have a probabilistic?avor to them,but formally they are not taken to be probabilistic.This is for good reason:should we interpret them as probabilities,results would be inconsistent with probabil-ity theory.Heckerman[10]and Lucas[11]discuss situations in which certainty factor computations can and cannot be correctly interpreted probabilistically. One reason they cannot is the incorrect treatment of bidirectional inference:two certainty factor rules(A←B):c1and B:c2imply nothing about inference from A to B,while P(A|B)and P(B)do place constraints on P(B|A).These problems are further discussed in Pearl[3].

12.2Probabilistic Logic Semantics

The semantic limitations of certainty factors is one of the motivations for de?ning precise semantics for probabilistic logics,but such investigations date from at least as far back as Carnap[12].

One of the most in?uential AI works in this regard is Nilsson[4](a similar approach is given by Hailperin[13]).Nilsson establishes a systematic way of determining the probabilities of logic sentences in a query set,given the proba-bilities of logical sentences in an evidence set.To be more precise,the method determines intervals of probabilities to the query sentences,since in principle the evidence set may be consistent with an entire range of point probabilities for them.For example,knowing that A is true with probability0.2and B with probability0.6means that A∧B is true with probability in[0,0.2],depending on whether A and B are mutually exclusive,or A→B,or anything in between.

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Given a set of sentences L,Nilsson considers the equivalence classes of possible worlds that assign the same truth values to the sentences in L(that is,as far as L is concerned,all possible worlds in the same class are the same).Formally, Nilsson’s system is based on the following linear problem:

Π=V P

0≤Πj≤1

0≤P i≤1

P i=1

i

whereΠis the vector of probabilities of sentences in both query and evidence sets,P the vector of probabilities of possible worlds equivalence classes,and V is a matrix with V ij=1if sentence j is true in possible world set i,and0 otherwise.The probabilities of sentences in the knowledge base are incorporated as constraints in this system as well,and linear programming techniques can be used to determine the probability of novel sentences.However,as Nilsson points out,the problem becomes intractable even with a modest number of sentences, since all possible worlds equivalence classes need to be enumerated and this is an intractable problem.Therefore this framework cannot be directly used in practice.

Placing the probabilities on the possible worlds,as does Nilsson,makes it easy to express subjective probabilities such as“Tweety?ies with probability 0.9”(that is,the sum of probabilities of all possible worlds in which Tweety ?ies is0.9).However,probabilistic knowledge can also express statistical facts about the domain such as“90%of birds?y”(which says that,in each possible world,90%of birds?y).Bacchus[5]provides an elaborate probabilistic logic semantics that includes both types of probabilistic knowledge,making it possible to use both statements above,as well as statements mixing them,such as“There is a probability of0.8that90%of birds?y.”He also discusses the interplay between the two types,namely the question of when it is correct to use the fact that“90%of birds?y”in order to assume that“a randomly chosen bird?ies with probability0.9,”a topic that has both formal and philosophical aspects. Halpern[6]elaborates on the axiomatization of Bacchus,taking probabilities to be real numbers(Bacchus did not),and is often cited as a reference for this semantics with two types of probabilities.In subsequent work,the subjective type probability has been much more developed and used,and is also the type involved in propositional graphical models.

Fagin,Halpern,Meggido[14]present a logic to reason about probabilities, including their addition and multiplication by scalars.Other works discussing the semantics of probabilities on?rst-order structures are[15,16,17].

12.3Extensional Approaches

Somewhat parallel to the works on the semantics of probabilistic logic,a di?erent line of research proposed logic reasoning systems incorporating uncertainty in

12A Survey of First-Order Probabilistic Models293 the explicit form of probabilities(as opposed to certainty factors).These systems often stem from the?elds of logic programming and deductive databases,and ?t into the category described by[3]as extensional systems,that is,systems in which rules work as“procedural licenses”for a computation step instead of a constraint on possible probability distributions.Most of these systems operate on a collection of rules or clauses that propagate generalized truth values(typically, a value or interval in[0,1]).

Kiefer and Li[18]provide a probabilistic interpretation and a?xpoint seman-tics to Shapiro[9].W¨u thrich[19]elaborates on their work,taking into account partial dependencies between clauses.For example,if each of them atoms a,b and c has a prior probability of0.5and we have two rules p←a∧b and p←b∧c,Kiefer and Li will assume the rules independent and assign a prob-ability0.25+0.25?0.25?0.25=0.4375to p.W¨u thrich’s system,however, takes into account the fact that b is shared by the clauses and computes instead 0.25+0.25?0.53=0.375(that is,it avoids double counting of the case where the two rules?re at the same time,which occurs only when the three atoms are true at once).

One of the most in?uential works within the extensional approach is Ng and Subrahmanian[20].Here,a logic programming system uses generalized truth values in the form of intervals of probabilities.They de?ne probabilistic logic program as sets of p-clauses of the form

A:μ←F1:μ1∧...F n:μn,

where A is an atom,F1,...,F n are basic formulas(conjunctions or disjunctions) andμ,μ1,...,μn are probability intervals.A clause states that if the probability of each formula F i is inμi,then the probability of A is inμ.For example,the clause

path(X,Y):[0.8,0.95]←a(X,Z):[1,1]∧path(Z,Y):[0.85,1]

states that,if a(X,Z)is certain(probability in interval[1,1]and therefore1) and path(Z,Y)has probability in[0.85,1],then path(X,Y)has probability in [0.8,0.95].Probabilities of basic formulas F i are determined from the probabil-ity intervals of their conjuncts(or disjuncts)by taking into account the possible correlations between them(similarly to what Nilsson does).The authors present a?xpoint semantics where clauses are repeatedly applied and probability inter-vals successively narrowed up to convergence.They also develop a model theory determining what models(sets of distributions on possible worlds)satisfy a probabilistic logic program,and a refutation procedure for querying a program.

Lakshmanan and Sadri[21]propose a system similar to Ngo and Subrahma-nian,while keeping track of both the probability of each atom as well of its negation.Additionally,it uses con?gurable independence assumptions for dif-ferent clauses,allowing the user to declare whether two atoms are independent, mutually exclusive,or even the lack of an assumption(as in Nilsson).Laksh-manan[22]separates the qualitative and quantitative aspects of probabilistic logic.Dependencies between atoms are declared in terms of the boolean truth

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values of a set of support atoms.Only later is a distribution assigned to the sup-port atoms,consequently de?ning distributions on the remaining atoms as well. The main advantage of the approach is the possibility of investigating di?erent total distributions,based on distributions on the support set,without having to recalculate the relationship between atoms and support set.The algorithms works in ways similar to Ngo and Haddawy[23]and Lakshmanan and Sadri[21], but propagates support set conditions rather than probabilities.Support sets are also a concept very similar to the hypotheses used in Probabilistic Abduction by Poole[24](see next section).

12.4Intensional Approaches

We now discuss intensional approaches to probabilistic logic languages,where statements(often in the form of rules)are interpreted as restrictions on a globally de?ned probability distribution.This probability distribution is over all possible worlds or,in other words,on assignments to the set of all possible random variables in the language.Statements typically pose constraints in the form of conditional probabilities,and often also as conditional independence relations. (As mentioned in Sect.12.2,another possibility would be statistical constraints, but this has not been explored in any works to our knowledge.) The algorithms in intensional approaches,when available,are arguably more complex than extensional approaches,since their steps do not directly correspond to the application of rules in the language and need to be consistent with the global distribution while being as local as possible(for e?ciency reasons).

We cover?ve di?erent types of intensional approaches:deduction rules,ex-haustive computation of derivations,sampling,Knowledge Based Model Con-struction(KBMC)and Lifted inference.

12.4.1Deduction Rules

Classical logic deduction systems often work by receiving a model speci?ed in a particular language and using deduction rules to derive new statements(guar-anteed to be true)from subsets of previous statements.Some work has been devoted to devising similar systems when the language is that of probabilistic logic.

This method is particularly challenging in probabilistic systems because prob-abilistic inference is not as modular as classical logical inference.For example, while the logical knowledge of A→B allows us to deduce B given that A∧?is true for any formula?,knowing P(B|A)in itself does not tell us anything about P(B|A∧?).In principle,one needs to consider all available knowledge when establishing the conditional probability of B.Classical logic reasoning shows a modularity that is harder to achieve in a probabilistic setting.

One way of making probabilistic inference more modular is to use knowledge about conditional independencies between random variables.If we know that B is independent of any other random variable given A,then we know that P(B|A∧?)

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is equal to P (B |A )for any ?.This has been the approach of graphical models such as Bayesian and Markov networks [3],where independencies are represented by the structure of a graph over the set of random variables.

The computation steps of speci?c inference algorithms for graphical models (such as Variable Elimination [25])could be cast as deduction rules,much like in classical logic.However this is not traditionally done,mostly because inference rules are typically described in a logic-like language and graphical models are not.When dealing with a ?rst-order probabilistic logic language,however,this approach becomes more natural.

Luckasiewicz [26]uses inference rules for solving trees of probabilistic con-ditional constraints over basic events.These trees are similar to Bayesian net-works,with each node being a random variable and each edge being labeled by a conditional probability table.However,these trees are not meant to encode in-dependence assumptions.Besides,conditional probabilities can also be speci?ed in intervals.

Frisch and Haddawy [27]present a set of inference rules for probabilistic propositional logic with interval probabilities.They characterize it as an anytime system since inference rules will increasingly narrow those intervals.They also provide more modular inference by allowing statements on conditional indepen-dencies of random variables,which are used by certain rules to derive statements based on local information.

Koller and Halpern [28]investigate the use of independence information for FOPI based on inference rules.They use this notion to discuss the issue of sub-stitution in probabilistic inference.While substitution is fundamental to classical logic inference,it is not sound in general in a probabilistic context.For example,

inferring P (q (A ))=13given ?P (q (X ))=13is not sound.Consider three pos-

sible worlds w 1,w 2,w 3containing the three objects o 1,o 2,o 3each,where q (o i )is 1in w i and 0otherwise.If each possible world has a probability 13of being

the actual world,then ?P (q (X ))=13holds.However,if A refers to o i in each

w i ,then P (q (A ))=1.While this problem can be solved by requiring constants to be rigid designators (that is,each of them refers to the same object in all worlds),the authors argue that this is too restrictive.Their solution is to use in-formation on independence.They show that when the statements ?P (q (X ))=13and x =A are independent,one can derive P (q (A ))=13.Finally,they discuss the topic of using statistical probabilities as a basis for subjective ones (the two types discussed by Bacchus [5]and Halpern [6])based on independencies.12.4.2Exhaustive Computation of Derivations

Another type of intensional system is the one in which the available algorithms exhaustively compute the set of derivations or proofs for a query,in the same way proofs are found for queries in logic programming.However,while in logic programming it is often only necessary to ?nd one proof for a certain query,in probabilistic models all proos will typically in?uence the query’s result,and therefore need to be computed.

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Riezler[29]presents a probabilistic account of Constraint Logic Programs (CLPs)[30].In regular logic programming,the only constraints over logical vari-ables are equational constraints coming from uni?cation.CLPs generalize this by allowing other constraints to be stated over those variables.These constraints are managed by special-purpose constraint solvers as the derivation proceeds,and failure in satisfying a constraint determines failure of the derivation.Probabilis-tic Constraint Logic Programs(PCLPs)are a stochastic generalization of CLPs, where clauses are annotated with a probability and chosen for the expansion of a literal according to that probability,among the available clauses with matching heads.The probability of a derivation is determined by the product of probabil-ities associated to the stochastic choices.In fact,PCLPs are a generalization of Stochastic Context-Free Grammars(SCFGs)[31],the di?erence between them being that PCLP symbols have arguments in the form of logical variables with associated constraints while grammar symbols do not.For this reason,PCLP derivations can fail while SCFGs will always succeed.This presents a compli-cation for PCLP algorithms because the probability has to be normalized with respect to the sum of successful derivations only.It also makes the use of ef-?cient dynamic programming techniques such as the inside-outside algorithm [32]not adequate for PCLPs,forcing us to compute all possible derivations of a query.Riezler focuses on presenting an algorithm for learning the parameters of a PCLP from incomplete data,in what is a generalization of the Baum-Welch algorithm for HMMs[33].

Stochastic Logic Programs[34,35]are very similar to PCLPs,restricting themselves to regular logic programming(e.g.,Prolog).This line of work is more focused on the development of an actual system on top of a Prolog interpreter and to be used with Inductive Logic Programming techniques such as Progol [36].Like Riezler,in[35]Cussens develops methods for learning parameters of SLPs using Improved Interative Scaling[37]and the EM algorithm[38].

Luckasiewicz[39]presents a form of Probabilistic Logic Programming that complements Nilsson’s[4]approach.Nilsson considers all equivalence classes of possible worlds with respect to the given knowledge and builds a linear pro-gram in order to assign probabilities to sentences.Luckasiewicz essentially does the same by using logic programming for both determining the the equivalence classes and the linear program.

Baral et al.[40]use answer set logic programming to implement a power-ful probabilistic logic language.Its distinguishing feature is the possibility of specifying observations and actions,with their corresponding implications with respect to causality,as studied by Pearl[41].However,the implementation,using answer set Prolog,depends on determining all answer sets.

12.4.3Sampling Approaches

Because building all drivations of a query given a program is very expensive, approximation solutions become an attractive alternative.

Sato[42]presents PRISM,a full-featured Prolog interpreter extended with probabilistic switches that can be used to encode probabilistic rules and facts.

12A Survey of First-Order Probabilistic Models297 These switches are special built-in predicates that randomly succeed or not, following a speci?c probability distribution.They can be placed in the body of a clause,which in consequence will succeed or not with the same distribution(when the rest of the body succeeds).Therefore,multiple executions of the program will yield di?erent random results that can be used as samples.A query can then be answered by multiple executions which sample its possible outcomes. Sato also provides a way of learning the parameters of the switches by using a form of the EM algorithm[43].

BLOG[44]is a?rst-order language allowing the speci?cation of a generative model on a?rst-order structure.It is similar in form to BUGS[45],a speci?cation language for propositional generative models.The main distinction of BLOG is its open world assumption;it does not require that the number of objects in the world be set a priori,using instead a prior on this number and also keeping track of identities of objects with di?erent names.BLOG computes queries by sampling over possible worlds.

12.4.4Knowledge Based Model Construction

We now present the most prominent family of models in the?eld of FOPI mod-els,Knowledge Based Model Construction(KBMC).These approaches work by generating a propositional graphical model from a?rst-order language speci?ca-tion that answers the query at hand.This construction is usually done in a way speci?c to the query,ruling out irrelevant portions of the graph so as to increase e?ciency.

Many KBMC approaches use a?rst-order logic-like speci?cation language,but some use di?erent languages such as frame systems,parameterized fragments of Bayesian networks,and description logics.Some build Bayesian networks while others prefer Markov networks(and in one case,Dependency Networks[46]).

While KBMC approaches try to prune sections of underlying graphical models which are irrelevant to the current query,there is still potentially much wasted computation because they may replicate portions of the graph which require es-sentially identical computations.For example,a problem may involve many em-ployees in a company,and the underlying graphical model will contain a distinct section with its own set of random variables for each of them(representing their properties),even though all these sections have essentially the same structure. Often the same computation will be repeated for each of those sections,while it is possible to perform it only once in a generalized form.Avoiding this waste is the object of Lifted First-Order Probabilistic Inference[47,48],discussed in Sect.12.4.5.

The most commonly referenced KBMC approach is that of Breese[49,50], although Horsch and Poole[51]had presented a similar solution a year before.

[49]de?nes a probabilistic logic programming language,with Horn clauses anno-tated by probabilistic dependencies between the clause’s head and body.Once a query is presented,clauses are applied to it in order to determine the prob-abilistic dependencies relevant to it.These dependencies are then used to form a Bayesian network.Backward inference will generate the causal portion of the

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network relative to the query;forward inference creates the diagnostic part.The

construction algorithm uses the evidence in order to decide when to stop expand-ing the network–there is no need to generate portions that are d-separated from the query by the evidence.In fact,this work covers not only Bayesian networks,

but in?uence diagrams as well,including decision and utility value nodes.

There are many works similar in spirit to[49]and di?ering only in some details;for example,the already mentioned Horsch and Poole[51],which also

uses mostly Horn clauses(it does allow for universal and existential quanti?ers over the entire clause body though)as the?rst-order language.One distinction

in this work,however,is the more explicit treatment of the issue of combination functions,used to combine distributions coming from distinct clauses with the same head.One example of a combination function is noisy-or[3],which assumes

that the probability provided by a single clause is the probability of it making the consequent true regardless of the other clauses.Suppose we have clauses A←B and A←C in the knowledge base,the?rst one dictating a probability

0.8for A when B is true and the second one dictating a probability0.7for

A when C is true.Then the combination function noisy-or builds a Conditional

Probability Table(CPT)with B and C as parents of A,with entries P(A|B,C)= {1?0.2×0.3,1?0.8×0.3,1?0.2×0.7,1?0.8×0.7}={0.94,0.76,0.86,0.47} for{(B= ,C= ),(B=⊥,C= ),(B= ,C=⊥),(B=⊥,C=⊥)},

respectively.

In?rst-order models,noisy-or and other combination functions are especially useful when a random variable has a varying number of parents,which makes its CPT impossible to represent by a?xed-dimensions table.A clause p←q(X),for example,determines that p depends on all instantiations of q(X),that is,all instantiations of q(X)are parents of p.However,the number of such instantiations depends on how many values X can take.Without knowing this number,the only way of having a general speci?cation of p’s CPT is to have a combination function on the instantiations of q(X).In fact,even when this number is known it may be convenient to represent the CPT with a combination function for compactness sake.

Charniak and Goldman[52]expand a deductive database and truth main-tenance system(TMS)in order to de?ne a language for constructing Bayesian networks.The Bayesian networks come from the data-dependency network main-tained by the TMS system,which is annotated with probabilities.There is also a notion of combination functions.The authors choose not to expand logical languages,justifying this choice by arguing that logic and probability do not correspond perfectly,the?rst being based on implication while the second on conditioning.

Poole[24]de?nes Probabilistic Abduction,a probabilistic logic language aimed at performing abduction reasoning.Probabilities are de?ned only for a set of predicates,called hypotheses(which is reminiscent of the support set in [22]),while the clauses themselves are deterministic.When a problem has nat-urally dependent hypotheses,one can rede?ne them as regular predicates and invent a new hypothesis to explain that dependence.While deterministic clauses

12A Survey of First-Order Probabilistic Models299 can seem too restrictive,one can always get the e?ect of probabilistic rules by using hypotheses as a condition of the rule(like switches in Sato’s PRISM[42]). The language also assumes that the bodies of clauses with the same head are mutually exclusive,and again this is not as restrictive as it might seem since clauses with non-mutually exclusive bodies can be rewritten as a di?erent set of clauses satisfying this.As in other works in this section,the actual computation of probabilities is based on the construction of a Bayesian network.In[53],Poole extends Probabilistic Abduction for decision theory,including both utility and decision variables,as well as negation as failure.

Glesner and Koller[54]present a Prolog-like language that allows the dec-laration of facts about a Bayesian network to be constructed by the inference process.The computing mechanisms of Prolog are used to de?ne the CPTs as well,so they are not restricted to tables,but can be computed on the?y.This allows CPTs to be de?ned as decision trees,for example,which provides a means of doing automatic pruning of the resulting Bayesian network–if the evidence provides information enough to make a CPT decision at a certain tree node, the descendants of that node,along with parts of the network relevant to those descendants only,do not need to be considered or built.The authors focus on ?exible dynamic Bayesian networks that do not necessarily have the same struc-ture at every time slice.

Haddawy[55]presents a language and construction method very similar to[51, 49].However,he focuses on de?ning the semantics of the?rst-order probabilistic logic language directly,and independently of the Bayesian network construction, and proceeds to use it to prove the correctness of the construction method.Breese [49]had done something similar by de?ning the semantics of the knowledge base as an abstract Bayesian network which does not usually get built itself in the presence of evidence,and by showing that the Bayesian network actually built will give the same result as the abstract one.

Koller and Pfe?er[56]present an algorithm for learning the probabilities of noisy?rst-order rules used for KBMC.They use the EM algorithm applied to the Bayesian networks generated by the model,using incomplete data.This works in the same way as the regular Bayesian network parameter learning with EM, with the di?erence that many of the parameters in the generated networks are in fact instances of the same parameter in a?rst-order rule.Therefore,all updates on these parameters must be accumulated in the original parameter.

Jaeger[57]de?nes a language for specifying a Bayesian network whose nodes are the extensions of?rst-order predicates.In other words,each node is the as-signment to the set all atoms of a certain predicate.Needless to say,inference in such a network would be extremely ine?cient since each node would have an extremely large number of values.However,it o?ers the advantage of mak-ing the semantics of the language very clear(it is just the usual propositional Bayesian network semantics–the extension of a predicate is just a propositional variable with a very large number of values).The author proposes,like other approaches here,to build a regular Bayesian network(with a random variable per ground atom)for the purpose of answering speci?c queries.He also presents

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a sophisticated scheme for combination functions,including the possibility of their nesting.

Koller et al.[58,59]de?ne Probabilistic Relational Models(PRMs),a sharp depart from the logical-probabilistic models that had been proposed until then as solutions for FOPI models.Instead of adding probabilities to some logic-like language,the authors use the formalism of Frame Systems[60]as a starting point.The language of frames,similar also to relational databases,is less expres-sive than?rst-order logic,which is to the authors one of its main advantages since?rst-order logic inference is known to be intractable(which only gets worse when probabilities are added to the mix).By using a language that limits its ex-pressivity to what is most needed in practical applications,one hopes to obtain more tractable inference,an argument commonly held in the Knowledge Rep-resentation community[61].In fact,Pfe?er and Koller had already investigated adding probabilities to restricted languages in[62].In that case,the language in question was that of description logics.

The language of Frame Systems consists of de?ning a set of objects described by attributes–binary predicates relating an object to a simple scalar value –or relations–binary predicates relating an object to another(or even self) object.PRMs add probabilities to frame systems by establishing distributions on attributes conditioned on other attributes(in the same object,or related object). In order to avoid declaring these dependencies for each object,this is done at a scheme level where classes,or template objects,stand for all instances of a class.This scheme describes the attributes of classes and the relations between them.Conditional probabilities are de?ned for attributes and can name the conditioning attributes via the relations needed to reach them.

As in the previous approaches,queries to PRMs are computed by generat-ing an underlying Bayesian network.Given a collection of objects(a database skeleton)and the relationships between them,a Bayesian network is built with a random variable for each attribute in each object.The parents of these random variables in the network are the ones determined by the relations in the particu-lar database,and the CPT?lled with the values speci?ed at the template level. An example of this process is shown in Fig.12.1.

Note that the set of ancestors of attributes in the underlying network is de-termined by the relations from one object to another.One could imagine an attribute rating of an object representing a restaurant that depends on the at-tribute training of the object representing its chef(related to it by the relation-ship chef).In approaches following?rst-order representations,chef would be a binary predicate,and each of its instances a random variable.As a result,the ancestors of rating would be the attributes training of all objects potentially linked to the restaurant by the relationship chef,plus the random variables standing for possible pairs in the relationship cook itself,resulting in a large (and thus expensive)CPT.PRMs avoid this when they take data with a de-?ned structure where the assignment to relations such as cook is known;in this case,the random variables in the relationship chef would not even be included in the network,and the attribute rating of each object would have a single

12A Survey of First-Order Probabilistic Models 301

(c)

Fig.12.1.(a)A PRM scheme showing classes of objects (rectangles),probabilistic dependencies between their attributes (full arrows)and relationships (dashed arrows).(b)A database skeleton showing a collection of objects,their classes and relationships.(c)The corresponding generated Bayesian network.

ancestor.When relationships are not ?xed in advance,we have structural uncer-tainty ,which was addressed by the authors in [63].These papers have presented PRM learning of both parameters and structure (that is,the learning of the scheme level).

PRMs make use of Bayesian networks,a directed graphical model that brings a notion of causality.In relational domains it is often the case that random variables depend on each other without a clear notion of causality.Take for example a network of people linked by friendship relationships,with the at-tribute smoker for each person.We might want to state the ?rst-order causal relationship P (smoker (X )|friends (X,Y ),smoker (Y ))in such a model,but it would create cycles in the underlying Bayesian network (between each pair of smoker attributes such as smoker (john )and smoker (mary )).For this reason,Relational Markov Networks (RMNs)[64]recast PRMs so they generate undi-rected graphical models (Markov networks)instead of Bayesian networks.In RMNs,dependencies are stated as ?rst-order features that get instantiated into potential function on cliques of random variables,without a notion of causality or conditional probabilities.The disadvantage of it,however,is that learning in undirected graphical models is harder than in directed ones,involving a full inference step at each expectation step of the EM algorithm.

Relational Dependency Networks (RDNs)[65]provide yet another alterna-tive to this problem.They are the ?rst-order version of Dependency Networks

302R.de Salvo Braz,E.Amir,and D.Roth

(DNs)(Heckerman,[46]),which use conditional probabilities but do not require https://www.wendangku.net/doc/5b4326586.html,ing directed conditional probabilities avoids the expensive learning of undirected models.However,DNs have the downside of conditional proba-bilities being no longer guaranteed consistent with the joint probability de?ned by their normalized product.Heckerman shows that,as the amount of training data increases,conditional probabilities in a DN will asymptotically converge to consistency.RDNs are sets of?rst-order conditional probabilities which are used to generate an underlying regular dependency network.These?rst-order conditional probabilities are typically learned from data by relational learners (Sect.12.5).RDNs are implemented in Proximity,a well-developed,publicly available software package.

Kersting and DeRaedt[66]introduce Bayesian Logic Programs.This work’s motivation is to provide a language which is as syntactically and conceptually simple as possible while preserving the expressive power of works such as Ngo and Haddawy[23],Jaeger[57]and PRMs[58].According to the authors,this is necessary so one understands the relationship between all these approaches,and also the fundamental aspects of FOPI models.

Fierens at al[67]de?ne Logical Bayesian Networks(LBNs).LBNs are very similar to Bayesian Logic Programs,with the di?erence of having both random variables and deterministic logical literals in their language.A logic programming inference process is run for the construction of the Bayesian network,during which logical literals are used,but since they are not random variables,they are not included in the Bayesian network.This addresses the same issue of ?xed relationships discussed in the presentation of PRMs,that is,when a set of relationships is deterministically known,we can create random variable nodes in the Bayesian network with signi?cantly fewer ancestors.In the BLPs and LBNs framework,this is exempli?ed by a rule such as:

rating(X)←cook(X,Y),training(Y).

which has an associated probability,declaring that a restaurant X’s rating depends on their cook Y’s training.In Bayesian Logic Programs,the instan-tiations of cook(X,Y)are random variables(just like the instantiations of rating(X)and training(Y)).Therefore,since we do not know a priori which Y makes cook(timpone,Y)true,rating(timpone)depends on all instantiations of cook(timpone,Y)and training(Y)and has all of them as parents in the un-derlying Bayesian network.If in the domain at hand the information of cook is deterministic,then this would be wasteful.We could instead determine Y such that cook(timpone,Y),say Y=joe,and build the Bayesian network with only the relevant random variable training(joe)as parent of rating(timpone). This is precisely what LBNs do.In LBNs,one would de?ne cook as a deter-ministic literal that would be reasoned about,but not included in the Bayesian network as a random variable.This in fact is even more powerful than the PRMs

12A Survey of First-Order Probabilistic Models 303

approach since it deals even with the situation where relationships are not di-rectly given as data,but have to be reasoned about in a deterministic manner.Santos Costa et al.

[68]propose an elegant KBMC approach that smoothly leverages an already existing framework,Constraint Logic Programming (CLP).In regular logic programming,the only constraint over logical variables are equa-tional constraints coming from uni?cation.As explained in Sect.12.4.2,CLP programs generalize this by allowing other constraints to be stated over those variables.These constraints are managed by special-purpose constraint solvers as the derivation proceeds,and failure in satisfying a constraint determines failure of the derivation.The authors leverage CLP by developing a constraint solver on probabilistic constraints expressed as CPTs,and simply plug it into an already existing CLP system.The resulting system can also use available logic program-ming mechanisms in the CPT speci?cation,making it possible to calculate it dynamically,based on the context,rather than by ?xed tables.The probabilistic constraint solver uses a Bayesian network internally in order to solve the posed constraints,so this system is also using an underlying propositional Bayesian network for answering queries.Santos Costa et al.indicate [69]as the closest approach to theirs,with the di?erence that the latter keeps hard constraints on Bayesian variables separate from probabilistic constraints.This allows hard constraints to be solved separately.It is also di?erent in that it does not use con-ditional independencies (like Bayesian networks do),and therefore its inference is exponential on the number of random variables.

Markov Logic Networks (MLNs)[70]is a recent and rapidly evolving frame-work for probabilistic logic.Its main distinctions are that it is based on undi-rected models and has a very simple semantics while keeping the expressive

1.5? Smokes(X) ?Cancer(X)“Smoking causes cancer”

?X Smokes(X) ?Cancer(X)

Weight Clausal form English / First-Order Logic

1.1

? Friends(X,Y) ?Smokes(X) ?Smokes(Y)“If two people are friends either both

smoke or neither does”

?X ?Y Fr(X,Y) ?(Sm(X) ?Sm(Y))Fig.12.2.A ground Markov network generated from a Markov Logic Network for objects Anna (A)and Bob (B)(example presented in [70])

304R.de Salvo Braz,E.Amir,and D.Roth

Fig.12.3.An example of a MEBN,as shown in[72]

power of?rst-order logic.The downside to this is that its inference can become quite slow if complex constructs are present.

MLNs consist of a set of weighted?rst-order formulas and a universe of ob-jects.Its semantics is simply that of a Markov network whose features are the instantiations of all these formulas given the universe of objects.The potential of a feature is de?ned as the exponential of its weight in case it is true.Figure12.2 shows an example.

Formulas can be arbitrary?rst-order logic formulas,which are converted to clausal form for inference.Converting existentially quanti?ed formulas to clausal form usually involves Skolemization,which requires uninterpreted functions in the language.Since MLNs do not include such functions,existentially quanti?ed formulas are replaced by the disjunction of their groundings(this is possible because the domain is?nite).The great expressivity of MLNs allows them to easily subsume other proposed FOPI languages.They are also a generalization of?rst-order logic,to which they reduce when weights are in?nite.

Learning algorithms for MLNs have been presented from the beginning.Because learning in undirected models is hard,MLNs use the notion of pseudo-likelihood [71],an approximate but e?cient method.When data is incomplete,EM is used.

12A Survey of First-Order Probabilistic Models305 MLNs are a powerful language and framework accompanied by well-supported software(called Alchemy)and which has been applied to real domains.The drawback of its expressivity is potentially very large underlying networks(for example,when existential quanti?cation is used).

Laskey[72]presents multi-entity Bayesian networks(MEBNs),a?rst-order version of Bayesian networks,which rely on generalizing typical Bayesian net-work representations rather than a logic-like language.A MEBN is a collection of Bayesian network fragments involving parameterized random variables.As in the other approaches,the semantics of the model is the Bayesian network re-sulting from instantiating these fragments.Once they are instantiated,they are put together according to the random variables they share.A MEBN is shown in Fig.12.3.

Laskey’s language is indeed quite rich,allowing in?nite models,function sym-bols and distributions on the parameters of random variables themselves.The work focus on de?ning this language rather than on the actual implementation, which is based on instantiating a Bayesian network containing the parts rele-vant to the query at hand.It does not provide a detailed account of this process, which can be especially tricky in the case of in?nite models.

12.4.5Lifted Inference

One of the major di?culties in KBMC approaches is that they must proposi-tionalize the model in order to perform inference.This does not preserve the rich ?rst-order structure present in the original model;the propositionalized version does not indicate anymore that CPTs are instantiations of the same original one, or that random variables are instantiations of an original parameterized random variable.In other words,it creates a potentially large propositional model with a great amount of redundancy that cannot be readily exploited.

Recent research on lifted inference[73,47,48]has addressed this point.A lifted inference algorithm receives a?rst-order speci?cation of a probabilistic model and performs inference directly on it,without propositionalization.This can potentially yield an enormous gain in e?ciency.

For example,a possible model can be formed by parameterized factors (or parfactors)φ1(epidemic(D))andφ2(sick(P,D))and a set of typed ob-jects flu,rubella,and john,mary etc.The model is equivalent to a propo-sitional graphical model formed by all possible instantiations of parfactors by the given objects,which is the set of regular factorsφ1(epidemic(flu)),φ1(epidemic(rubella)),...,andφ2(sick(john,flu)),φ2(sick(john,rubella)),φ2(sick(mary,flu)),

φ2(sick(mary,rubella)),etc.

What lifted inference does,instead of actually generating these instantiations, is to operate directly on the parfactors and obtain the same answer as the one obtained by instantiating and solving by a propositional algorithm.By operat-ing directly on parfactors,the lifted algorithm can potentially be much more e?cient,since the?rst-order structure is explicitly available to it.For example, suppose we want to compute the marginal of P(epidemic(flu)).Then we have

306R.de Salvo Braz,E.Amir,and D.Roth

to sum out all the other random variables in the model.While a regular KBMC algorithm would instantiate them and then sum them out,a lifted inference al-gorithm will directly sum out the parameterized epidemic (D ),for D =flu ,and sick (P,D ).The lifted elimination operation may not depend on the number of objects in the domain at all,greatly speeding up the process.The step in which an entire class of random variables is eliminated at once is possible because they all share the same structure,and this structure is explicitly available to the

algorithm.Figure 12.4presents a simpli?ed diagram of a lifted inference operation.

Poole [73]proposes a lifted algorithm that generalizes propositional Variable Elimination [25]but covers only some speci?c cases.de Salvo Braz et al.[47,48]present a broader algorithm,called First-Order Variable Elimination (FOVE).FOVE includes Poole’s operation (which this work calls Inversion Elimination )a generalized version of it,called simply Inversion ,and a second elimination operation called Counting Elimination .While Inversion does not depend on the domain size,Counting Elimination does,but still only exponentially less than propositionalization.The work also presents rigorous proofs of the correctness of these operations and shows how to solve the lifted version of the Most Probable Explanation (MPE)problem.While more general,FOVE still does not cover all possible cases,when it too must resort to propositionalization.When this

sick(P, D)Fig.12.4.A diagram of several possible operations involving ?rst-order and propo-sitional probabilistic models.The ?gure uses the notation of factor graphs,which ex-plictly shows potential functions as squares connected to their arguments.Parameter-ized factors are shown as piled up squares,since they compactly stand for multiple factors.Lifted inference operates solely on the ?rst-order representation and can be much faster than propositional inference,while producing the same results.

12A Survey of First-Order Probabilistic Models307 happens,this propositionalization will be localized to the parfactors involved in the uncovered case.

Lifted FOPI is a further step towards closing the gap between logic and prob-abilistic inference,bringing to the latter the type of inference that does not require binding of parameters(which would be the logical variables in atoms,in logical terms),often seen in the former.However,it has its own disadvantages. It is relatively more complicated to implement,and requires a normalizing pre-processing of the model(called shattering)that can be very expensive.Further methods are being developed to circumvent these di?culties.

12.5Relational Learning

In this section we discuss some?rst-order models developed from a machine learning perspective.

Machine learning algorithms have traditionally been de?ned as classi?cation of attribute-value vectors[74].In many applications,it is more natural and con-venient to represent data as graphs,where each vertex represents an object and each edge represents a relation between objects.Vertices can be labeled with at-tributes of its corresponding object(unary predicates or binary predicates where the second argument is a simple value;this is similar to PRMs in Sect.12.4.4), and edges can be labeled(the label can be interpreted as a binary predicate holding between the objects).This provides the typical data structure represen-tations such as trees,lists and collections of objects in general.When learning from graphs,we usually want to form hypotheses that explain one or more of the attributes and(or)relations(the targets)of objects in terms of its neigh-bors.Machine learning algorithms which were developed to bene?t from this type of representation have often been called relational.This is closely associ-ated to probabilistic?rst-order models,since graph data can be interpreted as a set of ground literals using unary and binary predicates.Because the hypothe-ses explaining target attributes and relations apply to several objects,it is also convenient to represent the learned hypotheses as quanti?ed(?rst-order)rules. And because most learners involve probabilities or at least some measure of un-certainty,probabilistic?rst-order rules provide a natural representation option. Figure12.5illustrates these concepts.

We now discuss three forms of relational learning:propositionalization(?at-tening),Inductive Logic Programming(ILP),and FOPI learning,which can be seen as a synthesis of the two.

12.5.1Propositionalization

A possible approach to relational machine learning is that of using a relational structure for generating propositional attribute-value vectors for each of its ob-jects.For this reason,the approach has been called propositionalization.Because it transforms graph-like data into vector-like data,it is also often called?attening.

Cumby&Roth[75]provide a language for transforming relational data into attribute-value vectors.Their concern is not forming a?rst-order hypothesis,

308R.de Salvo Braz,E.Amir,and D.Roth

Fig.12.5.A fragment of a graph structure used as input for relational learning.The same information can be represented as a set of ground literals(right).The hypotheses learned to explain either relations or attributes can be represented as weighted?rst-order clauses over those literals(below).

however.They instead keep the attribute-value hypothesis and transform novel data to that representation in order to classify it with propositional learners such as Perceptron.For example,in the case of Fig.12.5,a classi?er seeking to learn the relation acqt would go through the instances of that predicate and generate suitable attribute-value vectors.The literal acqt(paul,larry)would gen-erate an example with label acqt(X,Y)and features male(paul),male(larry), interest(paul,music),school(paul,fdr),interest(larry,basketball),

school(larry,fdr)etc,as well as non-ground ones such as male(X),male(Y), school(X,fdr),school(Y,fdr),school(X,Z),school(Y,Z),interest(X,music) etc.The literal acqt(joe,lissa)would generate an example with label acqt(X,Y) and features male(joe),female(lissa),interest(joe,baseball),

interest(joe,computers),school(joe,jfk),etc,as well as non-ground ones such as male(X),female(Y),school(X,jfk),school(Y,jfk),school(X,Z),

school(Y,Z)etc.Note how this reveals abstractions–the examples above share the features school(X,Z)and school(Y,Z),which may be one reason for people being acquaintances in this domain.Should a target depend on speci?c objects (say,it is much more likely for people at the FDR school to be acquainted to each other)not completely abstracted features such as school(X,fdr)would be preferred by the classi?er.

从实践的角度探讨在日语教学中多媒体课件的应用

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新视野大学英语全部课文原文

Unit1 Americans believe no one stands still. If you are not moving ahead, you are falling behind. This attitude results in a nation of people committed to researching, experimenting and exploring. Time is one of the two elements that Americans save carefully, the other being labor. "We are slaves to nothing but the clock,” it has been said. Time is treated as if it were something almost real. We budget it, save it, waste it, steal it, kill it, cut it, account for it; we also charge for it. It is a precious resource. Many people have a rather acute sense of the shortness of each lifetime. Once the sands have run out of a person’s hourglass, they cannot be replaced. We want every minute to count. A foreigner’s first impression of the U.S. is li kely to be that everyone is in a rush -- often under pressure. City people always appear to be hurrying to get where they are going, restlessly seeking attention in a store, or elbowing others as they try to complete their shopping. Racing through daytime meals is part of the pace

新视野大学英语第三版第二册课文语法讲解 Unit4

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