Probabilities of the Third Type: Statistical Relational Learning and Reasoning with Relative Frequencies

TL;DR

This paper introduces functional lifted Bayesian networks that model continuous dependencies on relative frequencies in relational data, improving probabilistic reasoning and learning across varying domain sizes.

Contribution

It presents a formalism for continuous frequency dependencies, compares it with existing models, and provides a scalable approach for parameter estimation in large domains.

Findings

Functional lifted Bayesian networks explicitly model continuous frequency dependencies.

The asymptotic probability distributions can be estimated from samples in large domains.

Convergence of parameters is uniform, ensuring reliable probabilistic modeling.

Abstract

Dependencies on the relative frequency of a state in the domain are common when modelling probabilistic dependencies on relational data. For instance, the likelihood of a school closure during an epidemic might depend on the proportion of infected pupils exceeding a threshold. Often, rather than depending on discrete thresholds, dependencies are continuous: for instance, the likelihood of any one mosquito bite transmitting an illness depends on the proportion of carrier mosquitoes. Current approaches usually only consider probabilities over possible worlds rather than over domain elements themselves. An exception are the recently introduced lifted Bayesian networks for conditional probability logic, which express discrete dependencies on probabilistic data. We introduce functional lifted Bayesian networks, a formalism that explicitly incorporates continuous dependencies on relative frequencies into statistical relational artificial intelligence, and compare and contrast them with lifted Bayesian networks for conditional probability logic. Incorporating relative frequencies is not only beneficial to modelling; it also provides a more rigorous approach to learning problems where training and test or application domains have different sizes. To this end, we provide a representation of the asymptotic probability distributions induced by functional lifted Bayesian networks on domains of increasing sizes. Since that representation has well-understood scaling behaviour across domain sizes, it can be used to estimate parameters for a large domain consistently from randomly sampled subpopulations. Furthermore, we show that in parametric families of FLBN, convergence is uniform in the parameters, which ensures a meaningful dependence of the asymptotic probabilities on the parameters of the model.