A Global Markov Property for Solutions of Stochastic Difference Equations and the corresponding Full Time Graphs

Tom Hochsprung, Jakob Runge, Andreas Gerhardus
Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence, PMLR 244:1698-1726, 2024.

Abstract

Structural Causal Models (SCMs) are an important tool in causal inference. They induce a graph and if the graph is acyclic, a unique observational distribution. A standard result states that in this acyclic case, the induced observational distribution satisfies a d-separation global Markov property relative to the induced graph. Time series can also be modelled like SCMs: One just interprets the stochastic difference equations that a time series solves as structural equations. However, technical problems arise when time series "start" at minus infinity. In particular, a d-separation global Markov property for time series and the corresponding infinite graphs, the so-called full time graphs, has thus far only been shown for stable vector autoregressive processes with independent finite-second-moment noise. In this paper, we prove a much more general version of this Markov property. We discuss our assumptions and study violations of them. Doing so hints at several pitfalls at the intersection of time series analysis and causal inference. Moreover, we introduce a new projection procedure for these infinite graphs which might be of independent interest.

Cite this Paper

BibTeX
@InProceedings{pmlr-v244-hochsprung24a, title = {A Global Markov Property for Solutions of Stochastic Difference Equations and the corresponding Full Time Graphs}, author = {Hochsprung, Tom and Runge, Jakob and Gerhardus, Andreas}, booktitle = {Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence}, pages = {1698--1726}, year = {2024}, editor = {Kiyavash, Negar and Mooij, Joris M.}, volume = {244}, series = {Proceedings of Machine Learning Research}, month = {15--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v244/main/assets/hochsprung24a/hochsprung24a.pdf}, url = {https://proceedings.mlr.press/v244/hochsprung24a.html}, abstract = {Structural Causal Models (SCMs) are an important tool in causal inference. They induce a graph and if the graph is acyclic, a unique observational distribution. A standard result states that in this acyclic case, the induced observational distribution satisfies a d-separation global Markov property relative to the induced graph. Time series can also be modelled like SCMs: One just interprets the stochastic difference equations that a time series solves as structural equations. However, technical problems arise when time series "start" at minus infinity. In particular, a d-separation global Markov property for time series and the corresponding infinite graphs, the so-called full time graphs, has thus far only been shown for stable vector autoregressive processes with independent finite-second-moment noise. In this paper, we prove a much more general version of this Markov property. We discuss our assumptions and study violations of them. Doing so hints at several pitfalls at the intersection of time series analysis and causal inference. Moreover, we introduce a new projection procedure for these infinite graphs which might be of independent interest.} }
Endnote
%0 Conference Paper %T A Global Markov Property for Solutions of Stochastic Difference Equations and the corresponding Full Time Graphs %A Tom Hochsprung %A Jakob Runge %A Andreas Gerhardus %B Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2024 %E Negar Kiyavash %E Joris M. Mooij %F pmlr-v244-hochsprung24a %I PMLR %P 1698--1726 %U https://proceedings.mlr.press/v244/hochsprung24a.html %V 244 %X Structural Causal Models (SCMs) are an important tool in causal inference. They induce a graph and if the graph is acyclic, a unique observational distribution. A standard result states that in this acyclic case, the induced observational distribution satisfies a d-separation global Markov property relative to the induced graph. Time series can also be modelled like SCMs: One just interprets the stochastic difference equations that a time series solves as structural equations. However, technical problems arise when time series "start" at minus infinity. In particular, a d-separation global Markov property for time series and the corresponding infinite graphs, the so-called full time graphs, has thus far only been shown for stable vector autoregressive processes with independent finite-second-moment noise. In this paper, we prove a much more general version of this Markov property. We discuss our assumptions and study violations of them. Doing so hints at several pitfalls at the intersection of time series analysis and causal inference. Moreover, we introduce a new projection procedure for these infinite graphs which might be of independent interest.
APA
Hochsprung, T., Runge, J. & Gerhardus, A.. (2024). A Global Markov Property for Solutions of Stochastic Difference Equations and the corresponding Full Time Graphs. Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 244:1698-1726 Available from https://proceedings.mlr.press/v244/hochsprung24a.html.

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