Resolution of Simpson's paradox via the common cause principle
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916330747199488 |
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| author | Hovhannisyan, A. Allahverdyan, A. E. |
| author_facet | Hovhannisyan, A. Allahverdyan, A. E. |
| contents | Simpson's paradox is an obstacle to establishing a probabilistic association between two events $a_1$ and $a_2$, given the third (lurking) random variable $B$. We focus on scenarios when the random variables $A$ (which combines $a_1$, $a_2$, and their complements) and $B$ have a common cause $C$ that need not be observed. Alternatively, we can assume that $C$ screens out $A$ from $B$. For such cases, the correct association between $a_1$ and $a_2$ is to be defined via conditioning over $C$. This setup generalizes the original Simpson's paradox: now its two contradicting options refer to two particular and different causes $C$. We show that if $B$ and $C$ are binary and $A$ is quaternary (the minimal and the most widespread situation for the Simpson's paradox), the conditioning over any binary common cause $C$ establishes the same direction of association between $a_1$ and $a_2$ as the conditioning over $B$ in the original formulation of the paradox. Thus, for the minimal common cause, one should choose the option of Simpson's paradox that assumes conditioning over $B$ and not its marginalization. The same conclusion is reached when Simpson's paradox is formulated via 3 continuous Gaussian variables: within the minimal formulation of the paradox (3 scalar continuous variables $A_1$, $A_2$, and $B$), one should choose the option with the conditioning over $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00957 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resolution of Simpson's paradox via the common cause principle Hovhannisyan, A. Allahverdyan, A. E. Methodology Artificial Intelligence Probability Data Analysis, Statistics and Probability Applications Simpson's paradox is an obstacle to establishing a probabilistic association between two events $a_1$ and $a_2$, given the third (lurking) random variable $B$. We focus on scenarios when the random variables $A$ (which combines $a_1$, $a_2$, and their complements) and $B$ have a common cause $C$ that need not be observed. Alternatively, we can assume that $C$ screens out $A$ from $B$. For such cases, the correct association between $a_1$ and $a_2$ is to be defined via conditioning over $C$. This setup generalizes the original Simpson's paradox: now its two contradicting options refer to two particular and different causes $C$. We show that if $B$ and $C$ are binary and $A$ is quaternary (the minimal and the most widespread situation for the Simpson's paradox), the conditioning over any binary common cause $C$ establishes the same direction of association between $a_1$ and $a_2$ as the conditioning over $B$ in the original formulation of the paradox. Thus, for the minimal common cause, one should choose the option of Simpson's paradox that assumes conditioning over $B$ and not its marginalization. The same conclusion is reached when Simpson's paradox is formulated via 3 continuous Gaussian variables: within the minimal formulation of the paradox (3 scalar continuous variables $A_1$, $A_2$, and $B$), one should choose the option with the conditioning over $B$. |
| title | Resolution of Simpson's paradox via the common cause principle |
| topic | Methodology Artificial Intelligence Probability Data Analysis, Statistics and Probability Applications |
| url | https://arxiv.org/abs/2403.00957 |