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 $c(y)=\tfrac12 y^2+\log m(y)$.

翻译：受复合决策问题中Tweedie公式的启发，我们研究了直接建模边际密度$m(y)$的经验贝叶斯估计量的理论基础。主要结果表明：在指数族模型中，$k \ge 3$次多项式对数边际不可能源于任何有效先验分布，而二次型恰好对应高斯先验。这从理论上解释了为何某些经验贝叶斯决策规则虽具有实用价值，却不对应任何形式的贝叶斯过程。我们通过证明边际密度仅当其在平滑参数邻域内可拓展为热方程的有界解时（超越$c(y)=\tfrac12 y^2+\log m(y)$的凸性条件）才构成高斯卷积，从而强化了该诊断判据。