# Partially Functional Dynamic Backdoor Diffusion-based Causal Model

arXiv: https://arxiv.org/abs/2509.00472 · HTML: https://mlsanity.com/paper/2509.00472/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 30 Aug 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 13 of 13)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Sample-Efficient Reinforcement Learning via Counterfactual-Based Data Augmentation](https://mlsanity.com/paper/2012.09092/) (2020)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Identifying Patient-Specific Root Causes with the Heteroscedastic Noise Model](https://mlsanity.com/paper/2205.13085/) (2022)
- [Modeling Causal Mechanisms with Diffusion Models for Interventional and Counterfactual Queries](https://mlsanity.com/paper/2302.00860/) (2023)
- [Counterfactual (Non-)identifiability of Learned Structural Causal Models](https://mlsanity.com/paper/2301.09031/) (2023)
- [Federated Adaptive Causal Estimation (FACE) of Target Treatment Effects](https://mlsanity.com/paper/2112.09313/) (2021)
- [DeCaFlow: A deconfounding causal generative model](https://mlsanity.com/paper/2503.15114/) (2025)
- [Diffusion Model in Causal Inference with Unmeasured Confounders](https://mlsanity.com/paper/2308.03669/) (2023)

## Abstract (arXiv)

Causal inference in spatio-temporal settings is critically hindered by unmeasured confounders with complex spatio-temporal dynamics and the prevalence of multi-resolution data. While diffusion models present a promising avenue for estimating structural causal models, existing approaches are limited by assumptions of causal sufficiency or static confounding, failing to capture the region-specific, temporally dependent nature of real-world latent variables or to directly handle functional variables. We bridge this gap by introducing the Partially Functional Dynamic Backdoor Diffusion-based Causal Model (PFD-BDCM), a unified generative framework designed to simultaneously tackle causal inference with dynamic confounding and functional data. Our approach formalizes a novel structural causal model that captures spatio-temporal dependencies in latent confounders through conditional autoregressive processes, represents functional variables via basis expansion coefficients treated as standard graph nodes, and integrates valid backdoor adjustment into a diffusion-based generative process. We provide theoretical guarantees on the preservation of causal effects under basis expansion and derive error bounds for counterfactual estimates. Experiments on synthetic data and a real-world air pollution case study demonstrate that PFD-BDCM outperforms existing methods across observational, interventional, and counterfactual queries. This work provides a rigorous and practical tool for robust causal inference in complex spatio-temporal systems characterized by non-stationarity and multi-resolution data.