# On the Conditional Independence Implication Problem: A Lattice-Theoretic Approach

arXiv: https://arxiv.org/abs/1408.2030 · HTML: https://mlsanity.com/paper/1408.2030/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2014.

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

- [Learning general conditional independence structures via the neighbourhood lattice](https://mlsanity.com/paper/2206.05829/) (2022)
- [Enabling Runtime Verification of Causal Discovery Algorithms with Automated Conditional Independence Reasoning (Extended Version)](https://mlsanity.com/paper/2309.05264/) (2023)

## Abstract (arXiv)

A lattice-theoretic framework is introduced that permits the study of the conditional independence (CI) implication problem relative to the class of discrete probability measures. Semi-lattices are associated with CI statements and a finite, sound and complete inference system relative to semi-lattice inclusions is presented. This system is shown to be (1) sound and complete for saturated CI statements, (2) complete for general CI statements, and (3) sound and complete for stable CI statements. These results yield a criterion that can be used to falsify instances of the implication problem and several heuristics are derived that approximate this "lattice-exclusion" criterion in polynomial time. Finally, we provide experimental results that relate our work to results obtained from other existing inference algorithms.