This paper presents a simple framework for Hornclause abduction, with probabilities associated with hypotheses. It is shown how this representation can represent any probabilistic knowledge representable in a Bayesian belief network. The main contributions are in finding a relationship between logical and probabilistic notions of evidential reasoning. This can be used as a basis for a new way to implement Bayesian Networks that allows for approximations to the value of the posterior probabilities, and also points to a way that Bayesian networks can be extended beyond a propositional language. 1

Traditional Machine Learning approaches based on single inference mechanisms have reached their limits. This causes the need for a framework that integrates approaches based on abduction and abstraction capabilities in the inductive learning paradigm, in the light of Michalski's Inferential Theory of Learning (ITL). This work is intended as a survey of the most significant contributions that are present in the literature, concerning single reasoning strategies and practical ways for bringing them together and making them cooperate in order to improve the effectiveness and efficiency of the learning process. The elicited role of an abductive proof procedure is tackling the problem of incomplete relevance in the incoming examples. Moreover, the employment of abstraction operators based on (direct and inverse) resolution to reduce the complexity of the learning problem is discussed. Lastly, a case study that implements the combined framework into a real multistrategy learning system is briefly presented.

Inference of abduction theories for handling

Abstract Statistical Relational Learning (SRL) is a growing field in Machine Learning that aims at the integration of logic-based learning approaches with probabilistic graphical models. Markov Logic Networks (MLNs) are one of the state-of-the-art SRL models that combine first-order logic and Markov networks (MNs) by attaching weights to first-order formulas and viewing these as templates for features of MNs. Learning models in SRL consists in learning the structure (logical clauses in MLNs) and the parameters (weights for each clause in MLNs). Structure learning of MLNs is performed by maximizing a likelihood function (or a function thereof) over relational databases and MLNs have been successfully applied to problems in relational and uncertain domains. However, most complex domains are characterized by incomplete data. Until now SRL models have mostly used Expectation-Maximization (EM) for learning statistical parameters under missing values. Multistrategic learning in the relational setting has been a successful approach to dealing with complex problems where multiple inference mechanisms can help solve different subproblems. Abduction is an inference strategy that has been proven useful for completing missing values in observations. In this paper we propose two frameworks for integrating abduction in SRL models. The first tightly integrates logical abduction with structure and parameter learning of MLNs in a single step. During structure search guided by conditional likelihood, clause evaluation is performed by first trying to logically abduce missing values in the data and then by learning optimal pseudo-likelihood parameters using the completed data. The second approach integrates abduction with Structural EM of [17] by performing logical abductive inference in the E-step and then by trying to maximize parameters in the M-step.