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Probabilistic inductive logic programming, sometimes also called statistical relational learning, addresses one of the central questions of artificial intelligence: the integration of probabilistic reasoning with first order logic representations and machine learning. A rich variety of different formalisms and learning techniques have been developed. In the present paper, we start from inductive logic programming and sketch how it can be extended with probabilistic methods. More precisely, we outline three classical settings for inductive logic programming, namely learning from entailment, learning from interpretations, and learning from proofs or traces, and show how they can be used to learn different types of probabilistic representations.

Bayesian networks provide an elegant formalism for representing and reasoning about uncertainty using probability theory. They are a probabilistic extension of propositional logic and, hence, inherit some of the limitations of propositional logic, such as the difficulties to represent objects and relations. We introduce a generalization of Bayesian networks, called Bayesian logic programs, to overcome these limitations. In order to represent objects and relations it combines Bayesian networks with definite clause logic by establishing a one-to-one mapping between ground atoms and random variables. We show that Bayesian logic programs combine the advantages of both definite clause logic and Bayesian networks. This includes the separation of quantitative and qualitative aspects of the model. Furthermore, Bayesian logic programs generalize both Bayesian networks as well as logic programs. So, many ideas developed in both areas carry over.Proposition 5.4. Let B be a Bayesian logic program, N its (possibly infinite) Bayesian network and X 1 , . . . , X m , m > 0, nodes of N . The support network N (X 1 , . . . , X m ) is the graph union G of all single support networks N (X i ).Proof. First, we show that N (X 1 , . . . , X m ) and G have the same set of nodes. The support network N (X 1 , . . . , X m ) has per definitionem a node A if and only if A is influencing a X i ∈ {X 1 , . . . , X m }. But, A is influencing a X i ∈ {X 1 , . . . , X m } if and only if A is a node in N (X i ), i.e. A is a node in G.Now, we prove that N (X 1 , . . . , X m ) and G have the same set of edges. A support network N (X 1 , . . . , X m ) has an edge E from a node A i to a node A j if and only if (1) both nodes, A i and A j are influencing a X k ∈ {X 1 , . . . , X m } and are therefore in N (X k ), and (2) the edge E is in N . Per definitionem of a support network this is if an only if the edge E is in N (X k ), i.e. E is an edge in G. 5.

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Although interactive learning puts the user into the loop, the learner remains mostly a black box for the user. Understanding the reasons behind predictions and queries is important when assessing how the learner works and, in turn, trust. Consequently, we propose the novel framework of explanatory interactive learning where, in each step, the learner explains its query to the user, and the user interacts by both answering the query and correcting the explanation. We demonstrate that this can boost the predictive and explanatory powers of, and the trust into, the learned model, using text (e.g. SVMs) and image classification (e.g. neural networks) experiments as well as a user study.