Abstract
We propose a formal language for describing and explaining statistical causality. Concretely, we define Statistical Causality Language (StaCL) for expressing causal effects and specifying the requirements for causal inference. StaCL incorporates modal operators for interventions to express causal properties between probability distributions in different possible worlds in a Kripke model. We formalize axioms for probability distributions, interventions, and causal predicates using StaCL formulas. These axioms are expressive enough to derive the rules of Pearl’s do-calculus. Finally, we demonstrate by examples that StaCL can be used to specify and explain the correctness of statistical causal inference.
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Notes
- 1.
An undirected path in a causal diagram \( G _{w}\) is said to be d-separated by \(\boldsymbol{z}\) if it has either (a) a chain
s.t. \(v\in \boldsymbol{z}\), (b) a fork
s.t. \(v\in \boldsymbol{z}\), or (c) a collider
s.t. \(v\not \in \boldsymbol{z}\cup \texttt{ANC}(\boldsymbol{z})\). \(\boldsymbol{x}\) and \(\boldsymbol{y}\) are said to be d-separated by \(\boldsymbol{z}\) if all undirected paths between variables in \(\boldsymbol{x}\) and in \(\boldsymbol{z}\) are d-separated by \(\boldsymbol{z}\).
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Acknowledgements
We thank Kenji Fukumizu for providing helpful information on the literature on causal inference. The authors are supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST. Yusuke Kawamoto is supported by JST, PRESTO Grant Number JPMJPR2022, Japan, and by JSPS KAKENHI Grant Number 21K12028, Japan. Tetsuya Sato is supported by JSPS KAKENHI Grant Number 20K19775, Japan. Kohei Suenaga is supported by JST CREST Grant Number JPMJCR2012, Japan.
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Kawamoto, Y., Sato, T., Suenaga, K. (2023). Formalizing Statistical Causality via Modal Logic. In: Gaggl, S., Martinez, M.V., Ortiz, M. (eds) Logics in Artificial Intelligence. JELIA 2023. Lecture Notes in Computer Science(), vol 14281. Springer, Cham. https://doi.org/10.1007/978-3-031-43619-2_46
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