# Function-Valued Causal Influence in Nonlinear Time Series

arXiv: https://arxiv.org/abs/2605.26408 · HTML: https://mlsanity.com/paper/2605.26408/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 12 of 12)

- [Visualizing the Effects of Predictor Variables in Black Box Supervised Learning Models](https://mlsanity.com/paper/1612.08468/) (2016)
- [Causal Influence Detection for Improving Efficiency in Reinforcement Learning](https://mlsanity.com/paper/2106.03443/) (2021)
- [Scalable Causal Discovery with Score Matching](https://mlsanity.com/paper/2304.03382/) (2023)
- [CAUSE: Learning Granger Causality from Event Sequences using Attribution Methods](https://mlsanity.com/paper/2002.07906/) (2020)
- [Neural Additive Vector Autoregression Models for Causal Discovery in Time Series](https://mlsanity.com/paper/2010.09429/) (2020)
- [Causal Inference by Identification of Vector Autoregressive Processes with Hidden Components](https://mlsanity.com/paper/1411.3972/) (2014)
- [The Multilinear Structure of ReLU Networks](https://mlsanity.com/paper/1712.10132/) (2017)
- [Jacobian Regularizer-based Neural Granger Causality](https://mlsanity.com/paper/2405.08779/) (2024)
- [Causal Effect Identification in LiNGAM Models with Latent Confounders](https://mlsanity.com/paper/2406.02049/) (2024)
- [Greater than the parts: A review of the information decomposition approach to causal emergence](https://mlsanity.com/paper/2111.06518/) (2021)
- [From Causal Discovery to Dynamic Causal Inference in Neural Time Series](https://mlsanity.com/paper/2603.20980/) (2026)
- [Trajectory-Aware Reliability Modeling of Democratic Systems](https://mlsanity.com/paper/2604.20127/) (2026)

## Abstract (arXiv)

Causal discovery in time series is increasingly performed using nonlinear machine-learning models, yet the resulting causal relationships are almost always summarized by scalar edge scores. We argue that this practice obscures the true object learned by nonlinear autoregressive models: a state-dependent function whose effect varies across regimes, magnitudes, and contexts. We formalize function-valued causal influence for additive, contribution-decomposable architectures and show that scalar causal scores constitute a severe information bottleneck, conflating between-state variation with within-state residual noise. Using Neural Additive Vector Autoregression as a representative architecture, we introduce a practical framework based on Individual Conditional Expectation for estimating causal response functions directly from trained models. Through controlled synthetic experiments, we demonstrate that edges with indistinguishable scalar scores can exhibit qualitatively different functional behaviors, including monotonic, thresholded, saturating, and sign-changing effects. An applied case study on democratic development further shows that function-valued analysis reveals regime-specific and asymmetric causal structure systematically missed by score-centric approaches.