# Partially Functional Dynamic Backdoor Diffusion-based Causal Model

## TL;DR

This paper introduces PFD-BDCM, a novel diffusion-based causal model that effectively handles dynamic confounders and functional data in complex spatio-temporal systems, improving causal inference accuracy.

## Contribution

The paper develops a unified generative framework that captures spatio-temporal dependencies and functional variables, with theoretical guarantees and practical improvements over existing methods.

## Key findings

- Outperforms existing methods on synthetic and real-world data
- Provides theoretical guarantees on causal effect preservation
- Demonstrates effectiveness in observational, interventional, and counterfactual analyses

## Abstract

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.