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**11 - 13**of**13**### Unifying Probability and Logic for Learning

"... Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem head on. Uncertain knowledge can be modeled by using graded probabilities rather tha ..."

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Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem head on. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values, but so far a completely satisfactory integration of logic and probability has been lacking. In particular the inability of confirming universal hypotheses has plagued most if not all systems so far. We address this problem head on. The main technical problem to be discussed is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii)

### The Australian National University

"... Abstract This paper studies the problem of probabilistic inference and learning in a logic-based setting. We show how probability densities, being functions, can be represented and reasoned with naturally and directly in higher-order logic, an expres-sive formalism not unlike the (informal) everyday ..."

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Abstract This paper studies the problem of probabilistic inference and learning in a logic-based setting. We show how probability densities, being functions, can be represented and reasoned with naturally and directly in higher-order logic, an expres-sive formalism not unlike the (informal) everyday language of mathematics. We give efficient inference algorithms and illustrate the general approach with a di-verse collection of applications. The problem of acquiring probabilistic theories in the context of agent systems is also considered. 1 Introduction Complex computer applications, especially those needing the technology of artificial intelligence,require rich knowledge representation languages. Such applications typically involve structured knowledge for which some form of logic provides a suitable language for knowledge representationand reasoning. These applications also typically involve uncertainty for which probability theory is required. Thus it is natural to look for knowledge representation languages that combine theexpressive power of both logic and probability.