# STOAT: Spatial-Temporal Probabilistic Causal Inference Network

arXiv: https://arxiv.org/abs/2506.09544 · HTML: https://mlsanity.com/paper/2506.09544/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 43 references that are in the map.
- First posted on arXiv 11 Jun 2025.

## Builds on (most cited first, 8 of 8)

- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes](https://mlsanity.com/paper/1910.03002/) (2019)
- [Learning Counterfactual Representations for Estimating Individual Dose-Response Curves](https://mlsanity.com/paper/1902.00981/) (2019)
- [Deep Factors for Forecasting](https://mlsanity.com/paper/1905.12417/) (2019)
- [Causal Inference for Time series Analysis: Problems, Methods and Evaluation](https://mlsanity.com/paper/2102.05829/) (2021)
- [The State of Applied Econometrics - Causality and Policy Evaluation](https://mlsanity.com/paper/1607.00699/) (2016)
- [Universal Difference-in-Differences for Causal Inference in Epidemiology](https://mlsanity.com/paper/2302.00840/) (2023)
- [Genomic-Informed Heterogeneous Graph Learning for Spatiotemporal Avian Influenza Outbreak Forecasting](https://mlsanity.com/paper/2505.22692/) (2025)

## Built on by (most cited first, 1 of 1)

- [DeepL'evy: Learning Heavy-Tailed Uncertainty in Highly Volatile Time Series](https://mlsanity.com/paper/2605.10364/) (2026)

## Abstract (arXiv)

Spatial-temporal causal time series (STC-TS) involve region-specific temporal observations driven by causally relevant covariates and interconnected across geographic or network-based spaces. Existing methods often model spatial and temporal dynamics independently and overlook causality-driven probabilistic forecasting, limiting their predictive power. To address this, we propose STOAT (Spatial-Temporal Probabilistic Causal Inference Network), a novel framework for probabilistic forecasting in STC-TS. The proposed method extends a causal inference approach by incorporating a spatial relation matrix that encodes interregional dependencies (e.g. proximity or connectivity), enabling spatially informed causal effect estimation. The resulting latent series are processed by deep probabilistic models to estimate the parameters of the distributions, enabling calibrated uncertainty modeling. We further explore multiple output distributions (e.g., Gaussian, Student's-t, Laplace) to capture region-specific variability. Experiments on COVID-19 data across six countries demonstrate that STOAT outperforms state-of-the-art probabilistic forecasting models (DeepAR, DeepVAR, Deep State Space Model, etc.) in key metrics, particularly in regions with strong spatial dependencies. By bridging causal inference and geospatial probabilistic forecasting, STOAT offers a generalizable framework for complex spatial-temporal tasks, such as epidemic management.