# On Projectivity in Markov Logic Networks

arXiv: https://arxiv.org/abs/2204.04009 · HTML: https://mlsanity.com/paper/2204.04009/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 8 Apr 2022.

## Builds on (most cited first, 9 of 9)

- [Symmetric Weighted First-Order Model Counting](https://mlsanity.com/paper/1412.1505/) (2014)
- [Weighted model counting beyond two-variable logic](https://mlsanity.com/paper/1804.10185/) (2018)
- [Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis](https://mlsanity.com/paper/2004.14724/) (2020)
- [An asymptotic analysis of probabilistic logic programming, with implications for expressing projective families of distributions](https://mlsanity.com/paper/2102.08777/) (2021)
- [Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula](https://mlsanity.com/paper/2110.05992/) (2021)
- [Inference, Learning, and Population Size: Projectivity for SRL Models](https://mlsanity.com/paper/1807.00564/) (2018)
- [Relational Marginal Problems: Theory and Estimation](https://mlsanity.com/paper/1709.05825/) (2017)
- [Dynamic group testing to control and monitor disease progression in a population](https://mlsanity.com/paper/2106.10765/) (2021)
- [Lifted Weight Learning of Markov Logic Networks Revisited](https://mlsanity.com/paper/1903.03099/) (2019)

## Built on by (most cited first, 5 of 5)

- [Lifted Inference beyond First-Order Logic](https://mlsanity.com/paper/2308.11738/) (2023)
- [Projectivity revisited](https://mlsanity.com/paper/2207.00625/) (2022)
- [Weighted First Order Model Counting with Directed Acyclic Graph Axioms](https://mlsanity.com/paper/2302.09830/) (2023)
- [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)

Markov Logic Networks (MLNs) define a probability distribution on relational structures over varying domain sizes. Many works have noticed that MLNs, like many other relational models, do not admit consistent marginal inference over varying domain sizes. Furthermore, MLNs learnt on a certain domain do not generalize to new domains of varied sizes. In recent works, connections have emerged between domain size dependence, lifted inference and learning from sub-sampled domains. The central idea to these works is the notion of projectivity. The probability distributions ascribed by projective models render the marginal probabilities of sub-structures independent of the domain cardinality. Hence, projective models admit efficient marginal inference, removing any dependence on the domain size. Furthermore, projective models potentially allow efficient and consistent parameter learning from sub-sampled domains. In this paper, we characterize the necessary and sufficient conditions for a two-variable MLN to be projective. We then isolate a special model in this class of MLNs, namely Relational Block Model (RBM). We show that, in terms of data likelihood maximization, RBM is the best possible projective MLN in the two-variable fragment. Finally, we show that RBMs also admit consistent parameter learning over sub-sampled domains.