We define a language for representing context-sensitive probabilistic knowledge. A knowledge base consists of a set of universally quantified probability sentences that include context constraints, which allow inference to be focused on only the relevant portions of the probabilistic knowledge. We provide a declarative semantics for our language. We present a query answering procedure which takes a query Q and a set of evidence E and constructs a Bayesian network to compute P (QjE). The posterior probability is then computed using any of a number of Bayesian network inference algorithms. We use the declarative semantics to prove the query procedure sound and complete. We use concepts from logic programming to justify our approach. Keywords: reasoning under uncertainty, Bayesian networks, Probability model construction, logic programming Submitted to Theoretical Computer Science special issue on Uncertainty in Databases and Deductive Systems. This work was partially supported by NSF g...

We present a method for dynamically generating Bayesian networks from knowledge bases consisting of first-order probability logic sentences. We present a subset of probability logic sufficient for representing the class of Bayesian networks with discrete-valued nodes. We impose constraints on the form of the sentences that guarantee that the knowledge base contains all the probabilistic information necessary to generate a network. We define the concept of d-separation for knowledge bases and prove that a knowledge base with independence conditions defined by d-separation is a complete specification of a probability distribution. We present a network generation algorithm that, given an inference problem in the form of a query Q and a set of evidence E, generates a network to compute P (QjE). We prove the algorithm to be correct. 1 Introduction The flexibility of Bayesian networks for representing probabilistic dependencies and the relative efficiency of computational techniques for p...

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Abstract. Current literature offers a number of different approaches to what could generally be called "probabilistic logic programming". These are usually based on Horn clauses. Here, we introduce a new formalism, Logic Programs with Annotated Disjunctions, based on disjunctive logic programs. In this formalism, each of the disjuncts in the head of a clause is annotated with a probability. Viewing such a set of probabilistic disjunctive clauses as a probabilistic disjunction of normal logic programs allows us to derive a possible world semantics, more precisely, a probability distribution on the set of all Herbrand interpretations. We demonstrate the strength of this formalism by some examples and compare it to related work.

The past few years have witnessed an significant interest in probabilistic logic learning, i.e. in research lying at the intersection of probabilistic reasoning, logical representations, and machine learning. A rich variety of di#erent formalisms and learning techniques have been developed. This paper provides an introductory survey and overview of the stateof -the-art in probabilistic logic learning through the identification of a number of important probabilistic, logical and learning concepts.

We present a probabilistic logic programming framework that allows the representation of conditional probabilities. While conditional probabilities are the most commonly used method for representing uncertainty in probabilistic expert systems, they have been largely neglected by work in quantitative logic programming. We define a fixpoint theory, declarative semantics, and proof procedure for the new class of probabilistic logic programs. Compared to other approaches to quantitative logic programming, we provide a true probabilistic framework with potential applications in probabilistic expert systems and decision support systems. We also discuss the relationship between such programs and Bayesian networks, thus moving toward a unification of two major approaches to automated reasoning. To appear in Proceedings of the 1995 Asian Computing Science Conference, Pathumthani, Thailand, December 1995. This work was partially supported by NSF grant IRI-9509165. 1 Introduction Reasoning u...

We review Logical Bayesian Networks, a language for probabilistic logical modelling, and discuss its relation to Probabilistic Relational Models and Bayesian Logic Programs. 1 Probabilistic Logical Models Probabilistic logical models are models combining aspects of probability theory with aspects of Logic Programming, first-order logic or relational languages. Recently a variety of languages to describe such models has been introduced. For some languages techniques exist to learn such models from data. Two examples are Probabilistic Relational Models (PRMs) [4] and Bayesian Logic Programs (BLPs) [5]. These two languages are probably the most popular and well-known in the Relational Data Mining community. We introduce a new language, Logical Bayesian Networks (LBNs) [2], that is strongly related to PRMs and BLPs yet solves some of their problems with respect to knowledge representation (related to expressiveness and intuitiveness). PRMs, BLPs and LBNs all follow the principle of Knowledge Based Model Construction: they offer a language that can be used to specify general probabilistic logical knowledge and they provide a methodology to construct a propositional model based on this knowledge when given a specific

We propose statistical abduction as a rst-order logical framework for representing, inferring and learning probabilistic knowledge. It semantically integrates logical abduction with a parameterized distribution over abducibles. We show that statistical abduction combined with tabulated search provides an e cient algorithm for probability computation, a Viterbi-like algorithm for nding the most likely explanation, and an EM learning algorithm (the graphical EM algorithm) for learning parameters associated with the distribution which achieve the same computational complexity as those specialized algorithms for HMMs (hidden Markov models), PCFGs (probabilistic context-free grammars) and sc-BNs (singly connected Bayesian networks).

We present a method for dynamically constructing Bayesian networks from knowledge bases consisting of first-order probability logic sentences. We present a subset of probability logic sufficient for representing the class of Bayesian networks with discretevalued nodes. We impose constraints on the form of the sentences that guarantee that the knowledge base contains all the probabilistic information necessary to construct a network. We define the concept of d-separation for knowledge bases and prove that a knowledge base with independence conditions defined by d-separation is a complete specification of a probability distribution. We present a network construction algorithm that, given an inference problem in the form of a query Q and a set of evidence E, constructs the smallest network to compute P (QjE). We prove the algorithm to be correct. Submitted to Journal of AI Research 1 Introduction The flexibility of Bayesian networks for representing probabilistic dependencies and the ...

Recursive loops in a logic program present a challenging problem to the PLP (Probabilistic Logic Programming) framework. On the one hand, they loop forever so that the PLP backward-chaining inferences would never stop. On the other hand, they may generate cyclic influences, which are disallowed in Bayesian networks. Therefore, in existing PLP approaches logic programs with recursive loops are considered to be problematic and thus are excluded. In this paper, we propose a novel solution to this problem by making use of recursive loops to build a stationary dynamic Bayesian network. We introduce a new PLP formalism, called a Bayesian knowledge base. It allows recursive loops and contains logic clauses of the form A ← A1,..., Al, true, Context, T ypes, which naturally formulates the knowledge that the Ais have direct influences on A in the context Context under the type constraints Types. We use the well-founded model of a logic program to define the direct influence relation and apply SLG-resolution to compute the space of random variables together with their parental connections. This establishes a clear declarative semantics for a Bayesian knowledge base. We view a logic program with recursive loops as a special temporal model, where backward-chaining cycles of the form A ←...A ←... are interpreted as feedbacks. This extends existing PLP approaches, which mainly aim at (non-temporal) relational models.