Probabilistic logic programming is a major part of statistical relational artificial intelligence, where approaches from logic and probability are brought together to reason about and learn from relational domains in a setting of uncertainty. However, the behaviour of statistical relational representations across variable domain sizes is complex, and scaling inference and learning to large domains remains a significant challenge. In recent years, connections have emerged between domain size dependence, lifted inference and learning from sampled subpopulations. The asymptotic behaviour of statistical relational representations has come under scrutiny, and projectivity was investigated as the strongest form of domain size dependence, in which query marginals are completely independent of the domain size. In this contribution we show that every probabilistic logic program under the distribution semantics is asymptotically equivalent to an acyclic probabilistic logic program consisting only of determinate clauses over probabilistic facts. We conclude that every probabilistic logic program inducing a projective family of distributions is in fact everywhere equivalent to a program from this fragment, and we investigate the consequences for the projective families of distributions expressible by probabilistic logic programs.
The behaviour of statistical relational representations across differently sized domains has become a focal area of research from both a modelling and a complexity viewpoint. In 2018, Jaeger and Schulte suggested projectivity of a family of distributions as a key property, ensuring that marginal inference is independent of the domain size. However, Jaeger and Schulte assume that the domain is characterised only by its size. This contribution extends the notion of projectivity from families of distributions indexed by domain size to functors taking extensional data from a database. This makes projectivity available for the large range of applications taking structured input. We transfer the known attractive properties of projective families of distributions to the new setting. Furthermore, we prove a correspondence between projectivity and distributions on countably infinite domains, which we use to unify and generalise earlier work on statistical relational representations in infinite domains. Finally, we use the extended notion of projectivity to define a further strengthening, which we call σ-projectivity, and which allows the use of the same representation in different modes while retaining projectivity.
Let σ be a first-order signature and let Wn be the set of all σ-structures with domain [n] = {1, . . . , n}. We can think of each structure in Wn as representing a "possible state of the world (or the context of interest)", or simply a "possible world" as we will say, following common informal terminology in the field of Statistical Relational Artificial Intelligence. By an inference framework we mean a class F of pairs (P, L), where P = (Pn : n = 1, 2, 3, . . .) and each Pn is a probability distribution on Wn, and L is a logic with truth values in the unit interval [0, 1].The inference frameworks that we consider will contain pairs (P, L) where P is determined by a so-called probabilistic graphical model, a concept used in AI and machine learning, and L is a logic with expressive capabilities of interest when analysing data sets, so for example, we consider logics which can express statements about, for example, (conditional) probabilities or (arithmetic or geometric) averages.From the point of view of probabilistic and logical expressivity one may consider an inference framework as optimal if it allows any pair (P, L) where P = (Pn : n = 1, 2, 3, . . .) is a sequence of probability distributions on Wn and L is a logic. But from the point of view of using a pair (P, L) from such an inference framework for making inferences on Wn when n is large we face the problem of computational complexity. This motivates looking for an "optimal" trade-off between expressivity and computational efficiency. The issue of computational complexity also arises when learning a probabilistic graphical model as one may want to use a formal language (logic) for describing events that are relevant for learning the graphical model. Learning a more complex graphical model, which in turn determines a sequence (Pn : n = 1, 2, 3, . . .) of more complex probability distributions on Wn, generally requires more computational resources.We define a notion that an inference framework is "asymptotically at least as expressive" as another inference framework. This relation is a preorder and we describe a (strict) partial order on the equivalence classes of some inference frameworks that in our opinion are natural in the context of machine learning and artificial intelligence, illustrated by Figure 1. The results have bearing on issues concerning efficient learning and probabilistic inference, but are also new instances of results in finite model theory about "almost sure elimination" of extra syntactic features (e.g quantifiers) beyond the connectives. Often such a result has a logical convergence law as a corollary.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.