D. Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in place of existentially quantified variables, or missing values, CLP(BN ) is most closely related to PRMs. The full paper also gives a detailed discussion of the relationship of CLP(BN ) to other probabilistic logics, including the work of Breese [2] Haddawy and Ngo [5] Sato [10] Poole [9], Koller and Pfeffer [7] Angelopolous [1] and Kersting and DeRaedt [6] To summarise that discussion in a sentence, CLP(BN ) does not replicate any of these approaches because they define probability distributions over sets of objects other than the set of ground Skolem terms. CLP(BN ) is a ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

....explanations for a query = q from a Horn CNF , deciding whether there is an additional explanation is NP complete. Consequently, the existence of a polynomial total time algorithm for computing all explanations implies P=NP. However, for the well known class of acyclic Horn theories (see e.g. [5, 24, 21, 1]) we present an algorithm which enumerates all explanations for q with incremental polynomial delay (i.e. in time polynomial in the size of the input and output so far) and thus solves the problem in polynomial total time. Compared to explanations for an atomic query q, intuitively cyclic ....

....literal query is infeasible in general unless P=NP, it becomes an issue to find restricted input classes for which this is feasible. In this section, we show a positive result for the important class of acyclic Horn theories, which has been studied extensively in the literature (see, e.g. [5, 24, 21, 1]) We first recall the concept of acyclic Horn theories (see e.g. 5, 24] Definition 2. For any Horn CNF over atom set At, its dependency graph is the directed graph G( V; E) where V = At and E = fx i x j j c 2 ; x i 2 N(c) x j 2 P (c)g, i.e. E contains an arc from each atom in a ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artif. Int., 64:81-130, 1993.

....query = q from a Horn CNF , deciding whether there is an additional explanation is NP complete. Consequently, the existence of a polynomial total time algorithm for INFSYS RR 1843 03 09 computing all explanations implies P=NP. However, for the well known class of acyclic Horn theories (see e.g. [5, 23, 20, 1]) we present an algorithm which enumerates all explanations for q with incremental polynomial delay (i.e. in time polynomial in the size of the input and output so far) and thus solves the problem in polynomial total time. Compared to explanations for an atomic query q, intuitively cyclic ....

....literal query is infeasible in general unless P=NP, it becomes an issue to find restricted input classes for which this is feasible. In this section, we show a positive result for the important class of acyclic Horn theories, which has been studied extensively in the literature (see, e.g. [5, 23, 20, 1]) We first recall the concept of acyclic Horn theories (see e.g. 5, 23] Definition 2 For any Horn CNF over atom set At, its dependency graph is the directed graph G( V; E) where V = At and E = fx i x j j c 2 ; x i 2 N(c) x j 2 P (c)g, i.e. E contains an arc from each atom in a ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--130, 1993.

....and learning algorithms. Second, downgrading does not focus on a particular PLM. Instead it systematically investigates the impact of language concepts. A general understanding of PLMs and learning PLMs is likely to emerge. PLM Probabilistic Formalism Logic Probabilistic Horn Abduction (PHA) [Poole, 1993] Bayesian Networks Prolog PRISM [Sato, 1995] Stochastic Grammars Prolog Stochastic Logic Programs (SLPs) Muggleton, 1996; Cussens, 2000] Stochastic Grammars Prolog Probabilistic Logic Programs (PLPs) Ngo and Haddawy, 1997] Bayesian Networks Prolog Bayesian Logic Programs (BLPs) Kersting and ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64:81--129, 1993.

....grid failure [6] Here abduction is used to abduce hypothetical physically possible events that might cause the diagnosis system to come up with a wrong diagnosis violating the specification constraints. Probabilistic Horn Abduction and Independence Choice Logic. Probabilistic Horn abduction [84], later extended into the independent choice logic [86] is a way to combine logical reasoning and belief networks into a simple and coherent framework. Its development has been motivated by the Theorist system [88] but it has been extended into a framework for decision and game theoretic agents ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

....base P satisfies an integrity constraint # IC iff P # is consistent . The first constraint is [p # p # ] where p # means the opposite of p, following the approach described in [8, 6] Belief revision in Bayesian networks can be accurately modeled by cost based abduction [12]. Polynomial algorithms exist for some useful classes of abductive problems [5] Since weighted abduction is not yet one of them, we are still exploring the best heuristics to use for our domain and application. This is known as the consistency view of integrity constraints (see for example ....

David Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

....in [7, 5] For example, if literal p stands for velocity of pumpkin is constant , then p is velocity of pumpkin is non constant . In other words, the abductive explanation satisfies this constraint iff Belief revision in Bayesian networks can be accurately modeled by cost based abduction [11]. Polynomial algorithms exist for some useful classes of abductive problems [6] Since weighted abduction is not yet one of them, we are still exploring the best heuristics to use for our domain and application. This is most closely related to the consistency view (see for example [7] ....

David Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

....context where different knowledge bases collaborate and compete in order to find a consistent diagnosis. The possibility of merging logical and probabilistic notions of evidential reasoning in unifying computational framework based on abduction has been the subject of a lot of work in literature [15, 16]. We consider here as reference the work of David Poole about probabilistic Horn Abduction [16] and show how its basic notions of probability and combination of probability can be easy introduced in the multi agent context of ALIAS. This allows us (under certain conditions) to generate di#erent ....

.... global sets of consistent hypothesis (representing the alternative diagnoses) each one characterized by a probability value that can be used for determining the best diagnosis. A further refinement could be achieved by taking into consideration Poole s work on Bayesian networks and abduction [15], since it allows to represent in an abductive framework not only probabilistically independent hypotheses, but also the probabilistic dependencies between symptoms and hypotheses. As a matter of future work we plan to study how to apply this approach in a multi agent context. We are aware that ....

D. L. Poole, Probabilistic Horn Abduction and Bayesian Networks. Artificial Intelligence, 64(1), 81-129, Elsevier, 1993.

....the above program for Tweety has legs, given that Tweety is a penguin is given by the uninformative interval . For this reason, many recent approaches towards integrating logic and probabilities combine logic based formalisms with Bayesian networks [60] In particular, Poole s work [63, 62] describes an approach to Horn clause abduction in which probabilities are associated with hypotheses. It is implemented by a generalization of SLD resolution. Haddawy and his group [22, 55] describe an approach to query processing in first order probabilistic knowledge bases by Bayesian network ....

....as the one produced by computing tight intervals under logical entailment in [45] We have used the principle of maximum entropy as a way to overcome the inferential weakness of model theoretic logical entailment. This approach has a number of advantages over the Bayesian network approaches in [63, 62, 22, 55, 24, 25]: Since the latter approaches originated from Bayesian networks, they all assume some strong structural restrictions on probabilistic knowledge bases. In particular, they all require that the grounding of a knowledge base is acyclic. Moreover, conditional probabilities are always given by a ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artif. Intell., 64:81--129, 1993.

....Koller and Pfe er s question, whether techniques from ILP could help to learn the logical component of rst order probabilistic models. 1 Introduction In recent years, there has been an increasing interest in integrating probability theory with rst order logic. One of the research streams [24, 22, 11, 6, 14] aims at integrating two powerful and popular knowledge representation frameworks: Bayesian networks [23] and rst order logic. In 1997, Koller and Pfe er [16] address the question where do the numbers come from for such frameworks. At the end of the same paper, they raise the question whether ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Arti cial Intelligence, 64:81-129, 1993.

....inference mechanisms for marginalizing them [81] Learning is typically applied to construct a probabilistic domain theory, e.g. the Bayes network, from examples. Recognizing the analogy, some researchers proposed methods that bridge the gap between logic and probabilistic representations [40, 36, 52, 84]. CES differs from Bayes networks in about the same way as C differs from PROLOG. Bayes networks specify joint distributions of random variables in a way that facilitates computationally efficient marginalization. Thus, inference mechanisms for Bayes networks keep track of all dependencies ....

D. Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64:81--129, 1993.

....component can be learned using a gradient based maximum likelihood method. 1 Introduction In recent years, there has been an increasing interest in integrating probability theory with rst order logic leading to di erent types of rst order probabilistic logics . One of the streams [22, 20, 11, 15, 12] concentrates on rst order extensions of Bayesian networks [21] i.e. it aims at integrating two powerful and popular knowledge representation frameworks: Bayesian networks and rst order logic. When investigating the state of the art in this stream (cf. 20, 11, 15, 12] then there are two ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Arti cial Intelligence, 64:81-129, 1993.

....Keywords: Bayesian networks, rst order logic, learning from interpretations, parameter estimation, structural learning, gradient, EM 1. Introduction In recent years, there has been an increasing interest in integrating probability theory with rst order logic. One of the research streams [Poo93,NH97, Jae97,FGKP99,KD01] concentrates on rst order extensions of Bayesian networks [Pea91] The reason why this has attracted attention is, that even though Bayesian networks are one of the most important, ecient and elegant frameworks for representing and reasoning with probabilistic models, ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Arti cial Intelligence, 64:81-129, 1993.

....so on. From the conditional independencies, the joint distribution of all propositions is reduced to the product of the conditional probabilities and the marginal probabilities. For example, the following equation holds for the Bayesian network in Figure 5. 3 (This network example is cited from [Poole 93] 53 0.685 0.818 1.0 Figure 5.2: Learned hidden markov model P( Tampering, Fire, Alarm, Smoke, Leaving, Report) P(ReportlLeaving ) P(LeavinglAlarm ) P( Alarml Tampering , Fire) P( SmokelFire ) P( Fire) P( Tampering) tampering fire alarm leaving report Figure 5.3: An example of a ....

Poole, D., Probabilistic Horn abduction and Bayesian networks, Artificial Intelligence 63, pp.81-129, 1993.

....msw(tr(s0) 3, s0) msw(tr(sl) 3, sl) Fig. 1. The support raph th the oal kmm( a,b,a] Related works So far, many probabilistic extensions of logic programs have been proposed. We here mention some of related works briefly (In [11] we have also mentioned other related works such as [4, 7]) Muggleton s stochastic logic programs (SLPs) 5] combine probabilities with first order logic programs, but no mention is made about the parameter learning. In Riezler s probabilistic constraint logic pro gramming [12] and Cussens s loglinear models using SLPs [3] the probability distribution ....

Poole, D., Probabilistic Horn abduction and Bayesian networks, Artificial Intelligence, Vol.64, pp.81-129, 1993.

....possible to tell, by purely symbolic reasoning, what the best candidate is. It seems that we draw on, more or less inevitably, probability as a means for measuring plausibility of the candidate. In logic programming, we can see a considerable body of research works that make use of probabilities [7, 8, 9, 13]. The objective of this paper is to provide basic components for a unified symbolic statistical information processing system in the framework of logic programming. The first one is a semantic basis for probabilistic computation. The second one is a general learning schema for logic programs. The ....

....5 describes an experimental resuit with the learning algorithm. Section 6 is conclusion referring to related work. 2 Distribution semantics 2. 1 Preliminaries The relationship between logic and probability is quite an old subject and its investigation is inherently of interdisciplinary nature ([3, 6, 7, 9, 11, 13]) One of our purposes here is to show how to assign probabilities to all first order formulae containing V and 3 over an infinite Herbrand universe in such a way that the assignment satisfies Kolmogoroff s axioms for probability and causes no inconsistency. This seems required because, for ....

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Poole,D., Probabilistic Horn abduction and Bayesian networks, Artifi- cial Intelligence 64, pp81-129, 1993.

....FTA tends to estimate such aspects by means of prior information; Bayesian netowrks, on the other hand can naturally address this point, by providing the analyst with a more reliable analysis. 5 More specific techniques relying on the logical semantics of a BN in terms of Horn clauses [16, 17] can also be devised, by adopting abductive reasoning. 6 As in FTA this is obtained by multiplying the prior of the true value of each node in the cut set. DI A A IObus Inp AA In A Ch A Sig A DI B Tribus A B IObus DI C C IObus Inp AB Inp AC Tribus C Tribus B Inp BA Inp BB Inp BC Inp CA ....

D. Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64(1):81--129, 1994.

....dual program. The ABDUAL framework has been applied in medical diagnosis [25] reasoning about actions [2] solving inconsistencies in metaphorical reasoning [58] and diagnosis of power grid failure [4] 12 Probabilistic Horn Abduction and Independence Choice Logic Probabilistic Horn abduction [73], which later evolved into the independent choice logic [75] is a way to combine logical reasoning and belief networks into a simple and coherent framework. Its development has been motivated by the Theorist system [77] but it has been extended into a framework for decision and game theoretic ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Articial Intelligence, 64(1):81-129, 1993.

.... Work Since CES computes with probability distribution, it is immanently related to Bayes networks (BNs) 6] The relation between CES and BNs is best explained by an analogy: CES is to BNs as are procedural (or object oriented) programming languages to declarative ones (like Prolog, see also [7]) In BNs, the inference and the knowledge representation are strictly separated, as is the case in Prolog; whereas in CES program statements are computational, like in C and C . This characteristic has multiple ramifications. For example, Bayes networks (like knowledge bases) can be used for ....

D. Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64:81--129, 1993.

....as a means of quantifying the uncertainty in the facts and relations, though it is possible to extend the approach to other uncertainty handling formalisms and IEEE Transactions on Knowledge and Data Engineering, 8(3) 353 372. 27 representation languages. A similar approach is adopted by Poole [118, 117] who uses rst order horn clause logic to represent the variables in a belief network and the relationships between them, attaching an associated probability to each. The logical clauses are then used to deduce various facts such that the probability associated with the facts is the probability ....

....relational databases. However, he also showed that there were some di erences between the ideas. Finally, in a recent paper, Wong et al. 163] have shown that bayesian networks can be represented as relational databases. Rather as one might expect given the work discussed above by Poole [118, 117], and the close correspondance between predicates and the tuples in a relational table, it seems that if a probability distribution is given over a set of relational tables, it is possible to perform correct probabilistic inference using just the project and join operations that one would expect ....

Poole, D. (1993) Probabilistic horn abduction and Bayesian networks, Articial Intelligence, 64, 81-129.

.... of Friedman et al. 1999) the construction of probabilistic relational models (Ngo Haddawy 1997, Jaeger 1997, Koller Pfe er 1998) These systems have evolved out of earlier frameworks that were developed as speci cation languages for structurally uniform classes of Bayesian networks (Poole 1993, Breese 1992, Saotti Umkehrer 1994) Given a particular probabilistic query, a speci cation in such a language would serve as the blueprint for the automatic generation of a Bayesian network in which the probability of the query then is computed. This method has been called knowledge based ....

Poole, D. (1993), `Probabilistic horn abduction and bayesian networks', Articial Intelligence 64, 81-129.

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993. 41

....of the conditional probability of a variable given its parents. This has been in terms of either causal independencies [Heckerman and Breese, 1994; Zhang and Poole, 1996] or by exploiting finer grained contextual independencies inherent in stating the conditional probabilities in terms of rules [Poole, 1993] or trees This work was supported by Institute for Robotics and Intelligent Systems, Project IC 7 and Natural Sciences and Engineering Research Council of Canada Research Grant OGPOO44121. Thanks to Holger Hoos and Mike Horsch for comments. Boutilier et al. 1996] In this paper we show how ....

....function for given its parents. In the first, this rule simply means the conditional probability assertion: B s P e p C p A A w t w A :e S D z where :e B R TVU R p w . The second interpretation [Poole, 1993] is as a set of definite clauses, with noise terms in the body. The noise terms are atoms that are grouped into independent alternatives (disjoint sets) that correspond to random variables. In this interpretation the above rule is interpreted as the clause: s P uv sp t p A A ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

....Joint Conference on AI(IJCAI 95) Montreal, August, 1995 3. There is a clean way to integrate this with models of uncertainty (e.g. for noisy sensors and sloppy and unreliable actuators) The logic programs here are of the form that can be used within a probabilistic Horn abduction system [Poole, 1993] . One of the aims of this work is to produce a representation for robot behaviour that is both suitable for controlling a real robot and also can be the output of a decision theoretic planning system. 4. The logic programs form an executable specification of what an agent should do. Although they ....

..... 4. 1 Noisy sensors and actuators The above axiomatisation showed how to model partial information about the environment (the agent had very limited sensing ability) In this section we sketch a way to model noisy sensors and actuators using a continuous version of probabilistic Horn abduction [Poole, 1993; 1995] The general idea of probabilistic Horn abduction is that there is a probability distribution over possible world generated by unconditionally independent random variables. A logic program gives the consequences of the random choices for each world. Formally, a possible world selects one ....

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

....decisions in a logical representation. 1 Introduction This paper presents the Independent Choice Logic (ICL) a logic for modelling multiple agents under uncertainty. It s inspired by game theory [53; 32; 17] Bayesian networks [35; 7] influence diagrams [23; 22] probabilistic Horn abduction [36] , structured representations of Bayesian networks and Markov decision processes [5; 8; 7] agent modelling and dynamical systems [29; 57; 51; 48] and logical modelling of action and change [27; 50; 45] First we motivate ICL from a number of different perspectives, then show how it fits within ....

....parents, nor does it specify a representation for the conditional probability of a variable given its parents in the network. The conditional probabilities of variables given their parents are typically represented as tables, but can often be specified more compactly in terms of trees [7] or rules [36] . Rules are more compact than trees (unless the trees can have shared structure and redundant tests) in the sense that there are some functions where the tree representation is exponentially larger than the rule representation, but the converse doesn t hold 1 . 1 As we allow negation as ....

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

....space of independent choices and a logic program that gives the consequences of these choices. The choices can be made by nature (which has probabilities over the choices) or by purposive agents (who are trying to maximise their utility) The ICL extends the author s probabilistic Horn abduction [Poole, 1993b] to include negation as failure and multiple agents. In this paper we only consider the decision theoretic (single agent under uncertainty) case. For the no agent case (with probabilities over choices) the rules induce an independence equivalent to that of Bayesian networks. The rules also allow ....

....interesting in its own right as a mix of logic and decision game theory [Poole, 1995b] The meshing is also easily described in this framework in terms of explanations . The ICL also naturally has a way to include logical variables, and thus we allow for parametrizable influence diagrams (see [Poole, 1993b] for a description of the purely probabilistic case) 2 The Independent Choice Logic The Independent Choice Logic specifies a way to build possible worlds. Possible worlds are built from choosing propositions from independent alternatives, and then extending these total choices with a logic ....

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81-- 129, 1993.

....to post action variables. 2 The post action nodes have the usual matrices describing the probability of their values given the values of their parents, under action ] We assume that these conditional probability matrices are represented using a treestructure (or if then rules) as done in [16, 13]. This representation is exploited to great effect in [2] in the solution of completely observable MDPs. We will adopt the same ideas below. The tree representation of the matrix for variable [ is illustrated in Figure 1(a) illustrating how asymmetries are exploited. The tree associated with ....

.... current state estimate without explicitly incorporating history allows policies to be expressed independent of history our 3 Note that in all of these network representations, the network is not strictly necessary the conditional probability trees themselves allow determination of parents [13]. 10 decisions are contingent only on the current state of belief. However, we have in fact potentially made the specification of a policy much more difficult. The space of possible belief states Z is uncountably infinite. Computing, or even associating, an action choice with each belief state ....

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Art. Intel., 64(1):81-- 129, 1993.

....agent may not know whether the preconditions of an action hold, but, for example, may be sure enough to want to try the action. All of the uncertainty in our rules is relegated to independent choices as in the independent choice logic [Poole, 1995b] an extension of probabilistic Horn abduction [Poole, 1993] ) This allows for a clean separation of the completeness assumed by Reiter s solution to the frame problem and the uncertainty we need for decision theory. Before we describe the theory there are some design choices incorporated into the framework: In the deterministic case, the trajectory ....

.... treats a variable having a particular value as a proposition (not imposing any particular syntax) the syntactic restrictions and the semantic construction ensure that the values of a variable are mutually exclusive and covering, as well as that the variables are unconditionally independent (see [Poole, 1993]) Definition 3.3 If is a set of sets, a selector function on is a mapping Cut such that b for all b . The range of selector function , written is the set j b l . Definition 3.4 Given ICL SC theory mn 00 568 q r , for each selector function on n ....

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81-- 129, 1993.

....Poole [32] and D Ambrosio [4] have proposed back chaining search algorithms for Bayesian networks. None of these are nearly as efficient as the one presented here. Even if we consider finding the single most normal world, the algorithm here corresponds to forward chaining on definite clauses (see [33]) which can be done in linear time, but backward chaining has to search and takes potentially exponential time. This paper deliberately takes the extreme position of seeing how far we can get when we exploit the distributions and not the structure of the network. Hopefully this can shed light on ....

....oracles that determine the values that are unspecified in the state. These can be compared to the use of stuck at zero ( and stuck at one ( t m ) failure states. This invention of causal hypotheses can be done in general to produce exactly the worlds in Bayesian networks [33]. Note that because a possible world specifies the values of all variables, there is no difference between a possible world which is consistent with a formula u and one which entails u . That is, for possible world I , I u iff I C u (this can be easily proved by induction on the ....

D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993. 41

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D. Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

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Poole, D.: Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence 64 (1993) 81--129

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Poole, D.: 1993, `Probabilistic Horn Abduction and Bayesian Networks'. Artificial Intelligence 64(1), 81--129.

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Poole, D. "Probabilistic Horn abduction and Bayesian networks," Artificial Intelligence (64) 1993, pp. 81-129.

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64:81--129, 1993.

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Journal of Artificial Intelligence, vol. 64, 1993.

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1), 1993.

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D. Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64(1):81--129, 1993. 200

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Poole, D. 1993. Probabilistic horn abduction and bayesian networks. Artificial Intelligence 64:81-- 129.

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D. Poole (1993). Probabilistic Horn abduction and Bayesian networks. In Artificial Intelligence, 64(1):81-129.

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Poole, D.: Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence 64 (1993) 81--129

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64:81--129, 1993.

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D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64:81--129, 1993.

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David Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1):81--129, 1993.

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D. Poole. Probabilistic horn abduction and Bayesian networks. Arti cial Intelligence, 64(1):81-129, 1993.

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Poole D. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64(1) (1993) 81-129.

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#64# D. Poole. Probabilistic Horn Abduction and Bayesian Networks. Arti#cial Intelligence, 64:81#

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D. Poole, `Probabilistic Horn abduction and Bayesian networks', Artificial Intelligence, 64(1), 81--129 (1993).

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Poole, D. 1993b. Probabilistic horn abduction and Bayesian networks. Artificial Intelligence 64:81--129.

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