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The shortest such program implies the uncompressed stored variable more-likely causes the computed variable.<ref>Kailash Budhathoki and Jilles Vreeken "[http://eda.mmci.uni-saarland.de/pubs/2016/origo-budhathoki,vreeken.pdf Causal Inference by Compression]" 2016 IEEE 16th International Conference on Data Mining (ICDM)</ref><ref>{{Cite journal |doi = 10.1007/s10115-018-1286-7|title = Telling cause from effect by local and global regression|journal = Knowledge and Information Systems|year = 2018|last1 = Marx|first1 = Alexander|last2 = Vreeken|first2 = Jilles|volume=60|issue = 3|pages=1277–1305|doi-access = free}}</ref>

The shortest such program implies the uncompressed stored variable more-likely causes the computed variable.<ref>Kailash Budhathoki and Jilles Vreeken "[http://eda.mmci.uni-saarland.de/pubs/2016/origo-budhathoki,vreeken.pdf Causal Inference by Compression]" 2016 IEEE 16th International Conference on Data Mining (ICDM)</ref><ref>{{Cite journal |doi = 10.1007/s10115-018-1286-7|title = Telling cause from effect by local and global regression|journal = Knowledge and Information Systems|year = 2018|last1 = Marx|first1 = Alexander|last2 = Vreeken|first2 = Jilles|volume=60|issue = 3|pages=1277–1305|doi-access = free}}</ref>

最短的这种程序意味着，更有可能是未压缩的'''<font color='#32cd32>存储变量stored variable</font>'''导致了'''<font color='#32cd32>计算出的变量computed variable</font>'''。

最短的这种程序意味着，更有可能是未压缩的'''<font color='#ff8000>存储变量stored variable</font>'''导致了'''<font color='#ff8000>计算出的变量computed variable</font>'''。

 

Incorporate an independent noise term in the model to compare the evidences of the two directions.

Incorporate an independent noise term in the model to compare the evidences of the two directions.

Here are some of the noise models for the hypothesis Y → X with the noise E:

Here are some of the noise models for the hypothesis Y → X with the noise E:

下面是一些假设 y → x 有噪声 e 的噪声模型:

下面是一些支持 Y → X 假设且具有噪声 E 的噪声模型:

* Additive noise:<ref>Hoyer, Patrik O., et al. "[https://papers.nips.cc/paper/3548-nonlinear-causal-discovery-with-additive-noise-models.pdf Nonlinear causal discovery with additive noise models]." NIPS. Vol. 21. 2008.</ref> <math>Y = F(X)+E</math>

* Additive noise:<ref>Hoyer, Patrik O., et al. "[https://papers.nips.cc/paper/3548-nonlinear-causal-discovery-with-additive-noise-models.pdf Nonlinear causal discovery with additive noise models]." NIPS. Vol. 21. 2008.</ref> <math>Y = F(X)+E</math>

* Linear noise:<ref>{{cite journal | last1 = Shimizu | first1 = Shohei | display-authors = etal | year = 2011 | title = DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model | url = http://www.jmlr.org/papers/volume12/shimizu11a/shimizu11a.pdf | journal = The Journal of Machine Learning Research | volume = 12 | issue = | pages = 1225–1248 }}</ref> <math>Y = pX + qE</math>

* Linear noise:<ref>{{cite journal | last1 = Shimizu | first1 = Shohei | display-authors = etal | year = 2011 | title = DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model | url = http://www.jmlr.org/papers/volume12/shimizu11a/shimizu11a.pdf | journal = The Journal of Machine Learning Research | volume = 12 | issue = | pages = 1225–1248 }}</ref> <math>Y = pX + qE</math>

* Post-non-linear:<ref>Zhang, Kun, and Aapo Hyvärinen. "[https://arxiv.org/pdf/1205.2599 On the identifiability of the post-nonlinear causal model]." Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence. AUAI Press, 2009.</ref> <math>Y = G(F(X)+E)</math>

* Post-non-linear:<ref>Zhang, Kun, and Aapo Hyvärinen. "[https://arxiv.org/pdf/1205.2599 On the identifiability of the post-nonlinear causal model]." Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence. AUAI Press, 2009.</ref> <math>Y = G(F(X)+E)</math>

* Heteroskedastic noise: <math>Y = F(X)+E.G(X)</math>

* Heteroskedastic noise: <math>Y = F(X)+E.G(X)</math>

* Functional noise:<ref name="Mooij">Mooij, Joris M., et al. "[http://papers.nips.cc/paper/4173-probabilistic-latent-variable-models-for-distinguishing-between-cause-and-effect.pdf Probabilistic latent variable models for distinguishing between cause and effect]." NIPS. 2010.</ref> <math>Y = F(X,E)</math>

* Functional noise:<ref name="Mooij">Mooij, Joris M., et al. "[http://papers.nips.cc/paper/4173-probabilistic-latent-variable-models-for-distinguishing-between-cause-and-effect.pdf Probabilistic latent variable models for distinguishing between cause and effect]." NIPS. 2010.</ref> <math>Y = F(X,E)</math>

 

这些模型的共同假设是:

这些模型的共同假设是:

* There are no other causes of Y.

* There are no other causes of Y.

* X and E have no common causes.

* X and E have no common causes.

* Distribution of cause is independent from causal mechanisms.

* Distribution of cause is independent from causal mechanisms.

 

 

On an intuitive level, the idea is that the factorization of the joint distribution P(Cause, Effect) into  P(Cause)*P(Effect | Cause) typically yields models of lower total complexity than the factorization into  P(Effect)*P(Cause | Effect). Although the notion of “complexity” is intuitively appealing, it is not obvious how it should be precisely defined.<ref name="Mooij"/> A different family of methods attempt to discover causal "footprints" from large amounts of labeled data, and allow the prediction of more flexible causal relations.<ref>Lopez-Paz, David, et al. "[http://www.jmlr.org/proceedings/papers/v37/lopez-paz15.pdf Towards a learning theory of cause-effect inference]" ICML. 2015</ref>

On an intuitive level, the idea is that the factorization of the joint distribution P(Cause, Effect) into  P(Cause)*P(Effect | Cause) typically yields models of lower total complexity than the factorization into  P(Effect)*P(Cause | Effect). Although the notion of “complexity” is intuitively appealing, it is not obvious how it should be precisely defined.<ref name="Mooij"/> A different family of methods attempt to discover causal "footprints" from large amounts of labeled data, and allow the prediction of more flexible causal relations.<ref>Lopez-Paz, David, et al. "[http://www.jmlr.org/proceedings/papers/v37/lopez-paz15.pdf Towards a learning theory of cause-effect inference]" ICML. 2015</ref>

在直观的层面上，这个想法是联合分布 p (因果)到 p (因果) * p (效果 | 原因)的因式分解通常产生的模型的总复杂性低于因式分解到 p (效果) * p (因果)。尽管“复杂性”的概念在直觉上很吸引人，但是它应该如何精确定义却并不明显。

在直观的层面上，这个想法是联合分布 P(Cause, Effect) 到 P(Cause)*P(Effect | Cause)的因式分解通常产生的模型的总复杂性低于因式分解到 p (效果) * p (因果)。尽管“复杂性”的概念在直觉上很吸引人，但是它应该如何精确定义却并不明显。

 

== In statistics and economics ==

== In statistics and economics ==