Probabilities of the Third Type: Statistical Relational Learning and Reasoning with Relative Frequencies
Felix Weitkämper
TL;DR
This paper tackles reasoning with relative-frequency dependencies in statistical relational AI by introducing Functional Lifted Bayesian Networks (FLBN), a Halpern Type III formalism that models continuous dependencies on relative frequencies via functions f_R of first-order frequency terms. FLBN is contrasted with Lifted Bayesian Networks for Conditional Probability Logic (LBN-CPL), which encode discrete, threshold-based dependencies; the authors show FLBNs can express generalised linear models and other regression-style aggregations, while LBN-CPL handles discrete triggers. A central contribution is a rigorous asymptotic analysis proving that, under mild continuity conditions, FLBNs are asymptotically equivalent to quantifier-free LBNs and that the convergence is uniform in model parameters, enabling reliable transfer learning across domains of different sizes. The results support learning from subpopulations and extrapolating to larger domains, with implications for parameter estimation, model selection, and scalable SRL in applications such as epidemiology and public health policy. The work also situates FLBN within the broader landscape of SRL formalisms, providing guidance on leveraging projective limits for learning while retaining expressive power beyond discrete, threshold-based relationships.
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.