# Causal Density Functions

arXiv: https://arxiv.org/abs/2606.00754 · HTML: https://mlsanity.com/paper/2606.00754/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 19 references that are in the map.
- First posted on arXiv 30 May 2026.

## Builds on (most cited first, 6 of 6)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Permutation-based Causal Inference Algorithms with Interventions](https://mlsanity.com/paper/1705.10220/) (2017)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Intuitionistic j-Do-Calculus in Topos Causal Models](https://mlsanity.com/paper/2510.17944/) (2025)

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

- [Latent Confounded Causal Discovery via Lie Bracket Geometry](https://mlsanity.com/paper/2606.19610/) (2026)
- [Infinitesimal Causality](https://mlsanity.com/paper/2606.24621/) (2026)

## Abstract (arXiv)

We study the full density ratio between a specified intervention regime Pa and an observational regime P0, ρa=dPa/dP0, under the prerequisite Pall P0. We call the regime-indexed ratio a causal density function when Pa is an identified or directly observed interventional law. The underlying Radon-Nikodym derivative and the identity  mathbbEa[f(Z)] = mathbbE0!left[f(Z)ρa(Z)right]  are classical importance weighting, not new identification results. Our narrower question is whether retaining the entire pointwise ratio is useful as a reusable diagnostic across several downstream functionals. We evaluate a two-density plug-in baseline through held-out moment transport and overlap stress tests on synthetic and perturbation data. We also report a pairwise graph-scoring heuristic as a negative result: its F1 is (0.10) on a synthetic DAG, (0.12) on Sachs, and (0.33) on a multi-regime chain. These experiments do not establish an estimation advantage over direct density-ratio, inverse-probability, Riesz, or doubly robust methods; they instead delimit what the pointwise ratio target and the present plug-in estimator do and do not provide.