Jyotishka Datta, Nicholas G. Polson
arXiv 6 Sep 2025 · Mathematics — Statistics Theory
arXiv:2509.05823 · PDF · DOI · OpenAlex · Extracted main text
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 $c(y)=\tfrac12 y^2+\log m(y)$.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Efron, B (2011) Tweedie’s formula and selection bias | 1.000 | 14 | 3 | 100% |
| 2 | Koenker, R. & Mizera, I (2014) Convex optimization, shape constraints, compound decisions, and empirical bayes rules | 1.000 | 11 | 3 | 100% |
| 3 | Robbins, H (1956) An Empirical Bayes Approach to Statistics | 0.874 | 7 | 2 | 100% |
| 4 | Ritov, Y (2024) No need for an oracle: the nonparametric maximum likelihood decision in the compound decision problem is minimax | 0.811 | 4 | 2 | 100% |
| 5 | Ghosh, S., Ignatiadis, N., Koehler, F. & Lee, A (2025) Stein's unbiased risk estimate and hyvärinen's score matching | 0.737 | 3 | 2 | 100% |
| 6 | Jiang, W. & Zhang, C.-H (2009) General maximum likelihood empirical bayes estimation of normal means | 0.737 | 3 | 2 | 100% |
| 7 | Kiefer, J. & Wolfowitz, J (1956) Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters | 0.737 | 3 | 2 | 100% |
| 8 | Hirschman, I. I. & Widder, D. V (1955) Convolution transform | 0.737 | 3 | 2 | 100% |
| 9 | Efron, B (2012) Large-scale inference: empirical Bayes methods for estimation, testing, and prediction, vol. 1 | 0.644 | 2 | 2 | 100% |
| 10 | Robbins, H (1951) Asymptotically subminimax solutions of compound decision problems | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 48 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Compound decisions and empirical Bayes via Bayesian nonparametrics | 0.405 | 1 | 1 |