# Projectivity revisited

arXiv: https://arxiv.org/abs/2207.00625 · HTML: https://mlsanity.com/paper/2207.00625/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 1 Jul 2022.

## Builds on (most cited first, 8 of 8)

- [Bayesian Models of Graphs, Arrays and Other Exchangeable Random Structures](https://mlsanity.com/paper/1312.7857/) (2013)
- [Symmetric Weighted First-Order Model Counting](https://mlsanity.com/paper/1412.1505/) (2014)
- [An asymptotic analysis of probabilistic logic programming, with implications for expressing projective families of distributions](https://mlsanity.com/paper/2102.08777/) (2021)
- [Relational Bayesian Networks](https://mlsanity.com/paper/1302.1550/) (2013)
- [Inference, Learning, and Population Size: Projectivity for SRL Models](https://mlsanity.com/paper/1807.00564/) (2018)
- [A Complete Characterization of Projectivity for Statistical Relational Models](https://mlsanity.com/paper/2004.10984/) (2020)
- [On Projectivity in Markov Logic Networks](https://mlsanity.com/paper/2204.04009/) (2022)
- [Stochastic Planning and Lifted Inference](https://mlsanity.com/paper/1701.01048/) (2017)

## Built on by (most cited first, 2 of 2)

- [The generalised distribution semantics and projective families of distributions](https://mlsanity.com/paper/2211.06751/) (2022)
- [Understanding Domain-Size Generalization in Markov Logic Networks](https://mlsanity.com/paper/2403.15933/) (2024)

## Abstract (arXiv)

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.Recently, projectivity of a family of distributions emerged as a key property, ensuring that marginal probabilities are independent of the domain size. However, the formalisation used currently assumes 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 key known results on projective families of distributions to the new setting. This includes a characterisation of projective fragments in different statistical relational formalisms as well as a general representation theorem for projective families of distributions. 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.