Logarithmic Asymptotic Relations Between $p$-Values and Mutual Information
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| Format: | Preprint |
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2023
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| _version_ | 1866908799225298944 |
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| author | Mori, Tsutomu Kawamura, Takashi |
| author_facet | Mori, Tsutomu Kawamura, Takashi |
| contents | We establish a precise connection between statistical significance in dependence testing and information-theoretic dependence as quantified by Shannon mutual information (MI). In the absence of prior distributional information, we consider a maximum-entropy model and show that the probability associated with the realization of a given magnitude of MI takes an exponential form, yielding a corresponding tail-probability interpretation of a $p$-value. In contingency tables with fixed marginal frequencies, we analyze Fisher's exact test and prove that its $p$-value $P_F$ satisfies a logarithmic asymptotic relation of the form $MI=-(1/N)\log P_F + O(\log(N+1)/N)$ as the sample size $N\to\infty$. These results clarify the role of MI as the exponential rate governing the asymptotic behavior of $p$-values in the settings studied here, and they enable principled comparisons of dependence across datasets with different sample sizes. We further discuss implications for combining evidence across studies via meta-analysis, allowing mutual information and its statistical significance to be integrated in a unified framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_14735 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Logarithmic Asymptotic Relations Between $p$-Values and Mutual Information Mori, Tsutomu Kawamura, Takashi Statistics Theory 62E20 We establish a precise connection between statistical significance in dependence testing and information-theoretic dependence as quantified by Shannon mutual information (MI). In the absence of prior distributional information, we consider a maximum-entropy model and show that the probability associated with the realization of a given magnitude of MI takes an exponential form, yielding a corresponding tail-probability interpretation of a $p$-value. In contingency tables with fixed marginal frequencies, we analyze Fisher's exact test and prove that its $p$-value $P_F$ satisfies a logarithmic asymptotic relation of the form $MI=-(1/N)\log P_F + O(\log(N+1)/N)$ as the sample size $N\to\infty$. These results clarify the role of MI as the exponential rate governing the asymptotic behavior of $p$-values in the settings studied here, and they enable principled comparisons of dependence across datasets with different sample sizes. We further discuss implications for combining evidence across studies via meta-analysis, allowing mutual information and its statistical significance to be integrated in a unified framework. |
| title | Logarithmic Asymptotic Relations Between $p$-Values and Mutual Information |
| topic | Statistics Theory 62E20 |
| url | https://arxiv.org/abs/2308.14735 |