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 unit variance. Let $G_1$ be a possibly misspecified prior with zero mean and unit variance. We show that the squared error Bayes risk of the posterior mean under $G_1$ is bounded, uniformly over $G_0, G_1, \sigma^2 > 0$.

翻译：考虑正态位置模型 $X \mid \theta \sim N(\theta, \sigma^2)$，其中 $\sigma^2$ 已知。假设 $\theta \sim G_0$，且先验分布 $G_0$ 具有零均值和单位方差。令 $G_1$ 为可能误设的先验分布，也满足零均值和单位方差。我们证明，在 $G_1$ 下后验均值的平方误差贝叶斯风险在 $G_0$、$G_1$ 和 $\sigma^2 > 0$ 上一致有界。