An Asymptotic Analysis of Probabilistic Logic Programming, with Implications for Expressing Projective Families of Distributions

Over the last years, there has been increasing research on the scaling behaviour of statistical relational representations with the size of the domain, and on the connections between domain size dependence and lifted inference. In particular, the asymptotic behaviour of statistical relational representations has come under scrutiny, and projectivity was isolated as the strongest form of domain size independence. In this contribution we show that every probabilistic logic program under the distribution semantics is asymptotically equivalent to a probabilistic logic program consisting only of range-restricted clauses over probabilistic facts. To facilitate the application of classical results from finite model theory, we introduce the abstract distribution semantics, defined as an arbitrary logical theory over probabilistic facts to bridge the gap to the distribution semantics underlying probabilistic logic programming. In this representation, range-restricted logic programs correspond to quantifier-free theories, making asymptotic quantifier results avilable for use. We can conclude that every probabilistic logic program inducing a projective family of distributions is in fact captured by this class, and we can infer interesting consequences for the expressivity of probabilistic logic programs as well as for the asymptotic behaviour of probabilistic rules.Comment: 14 page

Labeled Directed Acyclic Graphs: a generalization of context-specific independence in directed graphical models

We introduce a novel class of labeled directed acyclic graph (LDAG) models for finite sets of discrete variables. LDAGs generalize earlier proposals for allowing local structures in the conditional probability distribution of a node, such that unrestricted label sets determine which edges can be deleted from the underlying directed acyclic graph (DAG) for a given context. Several properties of these models are derived, including a generalization of the concept of Markov equivalence classes. Efficient Bayesian learning of LDAGs is enabled by introducing an LDAG-based factorization of the Dirichlet prior for the model parameters, such that the marginal likelihood can be calculated analytically. In addition, we develop a novel prior distribution for the model structures that can appropriately penalize a model for its labeling complexity. A non-reversible Markov chain Monte Carlo algorithm combined with a greedy hill climbing approach is used for illustrating the useful properties of LDAG models for both real and synthetic data sets.Comment: 26 pages, 17 figure