Consider a normal location model $X \mid \theta \sim N(\theta, \sigma^2)$ with known $\sigma^2$. Suppose $\theta \sim G_0$, where the prior $G_0$ has zero mean and variance bounded by $V$. Let $G_1$ be a possibly misspecified prior with zero mean and variance bounded by $V$. We show that the squared error Bayes risk of the posterior mean under $G_1$ is bounded, subjected to an additional tail condition on $G_1$, uniformly over $G_0, G_1, \sigma^2 > 0$.

翻译：考虑已知方差 $\sigma^2$ 的正态位置模型 $X \mid \theta \sim N(\theta, \sigma^2)$。假设 $\theta \sim G_0$，其中先验 $G_0$ 具有零均值且方差以 $V$ 为界。令 $G_1$ 为一个可能设定错误的先验，其具有零均值且方差以 $V$ 为界。我们证明，在 $G_1$ 满足一个额外的尾部条件的前提下，$G_1$ 下后验均值的平方误差贝叶斯风险是有界的，且该界关于 $G_0, G_1, \sigma^2 > 0$ 是一致的。