This paper studies the functional uniqueness and stability of Gaussian priors in optimal $L^1$ estimation. While it is well known that the Gaussian prior uniquely induces linear conditional means under Gaussian noise, the analogous question for the conditional median (i.e., the optimal estimator under absolute-error loss) has only recently been settled. Building on the prior work establishing this uniqueness, we develop a quantitative stability theory that characterizes how approximate linearity of the optimal estimator constrains the prior distribution. For $L^2$ loss, we derive explicit rates showing that near-linearity of the conditional mean implies proximity of the prior to the Gaussian in the Lévy metric. For $L^1$ loss, we introduce a Hermite expansion framework and analyze the adjoint of the linearity-defining operator to show that the Gaussian remains the unique stable solution. Together, these results provide a more complete functional-analytic understanding of linearity and stability in Bayesian estimation under Gaussian noise.

翻译：本文研究了高斯先验在最优$L^1$估计中的函数唯一性与稳定性。尽管已知高斯先验在高斯噪声下唯一诱导线性条件均值，但关于条件中位数（即绝对误差损失下的最优估计量）的类似问题直至最近才得以解决。基于先前确立该唯一性的工作，我们发展了一种定量稳定性理论，刻画了最优估计量的近似线性如何约束先验分布。对于$L^2$损失，我们推导出显式速率，表明条件均值的近线性意味着先验在Lévy度量下接近高斯分布。对于$L^1$损失，我们引入埃尔米特展开框架，并通过分析线性定义算子的伴随算子，证明高斯分布仍是唯一稳定的解。这些结果共同为高斯噪声下贝叶斯估计中的线性与稳定性提供了更完备的函数分析理解。