Polynomial Log-Marginals and Tweedie's Formula : When Is Bayes Possible?
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917224327938048 |
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| author | Datta, Jyotishka Polson, Nicholas G. |
| author_facet | Datta, Jyotishka Polson, Nicholas G. |
| contents | Motivated by Tweedie's formula for the Compound Decision problem, we examine the theoretical foundations of empirical Bayes estimators that directly model the marginal density $m(y)$. Our main result shows that polynomial log-marginals of degree $k \ge 3 $ cannot arise from any valid prior distribution in exponential family models, while quadratic forms correspond exactly to Gaussian priors. This provides theoretical justification for why certain empirical Bayes decision rules, while practically useful, do not correspond to any formal Bayes procedures. We also strengthen the diagnostic by showing that a marginal is a Gaussian convolution only if it extends to a bounded solution of the heat equation in a neighborhood of the smoothing parameter, beyond the convexity of $c(y)=\tfrac12 y^2+\log m(y)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial Log-Marginals and Tweedie's Formula : When Is Bayes Possible? Datta, Jyotishka Polson, Nicholas G. Statistics Theory Econometrics Methodology 62C12, 60E10, 62C25 Motivated by Tweedie's formula for the Compound Decision problem, we examine the theoretical foundations of empirical Bayes estimators that directly model the marginal density $m(y)$. Our main result shows that polynomial log-marginals of degree $k \ge 3 $ cannot arise from any valid prior distribution in exponential family models, while quadratic forms correspond exactly to Gaussian priors. This provides theoretical justification for why certain empirical Bayes decision rules, while practically useful, do not correspond to any formal Bayes procedures. We also strengthen the diagnostic by showing that a marginal is a Gaussian convolution only if it extends to a bounded solution of the heat equation in a neighborhood of the smoothing parameter, beyond the convexity of $c(y)=\tfrac12 y^2+\log m(y)$. |
| title | Polynomial Log-Marginals and Tweedie's Formula : When Is Bayes Possible? |
| topic | Statistics Theory Econometrics Methodology 62C12, 60E10, 62C25 |
| url | https://arxiv.org/abs/2509.05823 |