Best sampling method within the normal family

by Regio   Last Updated July 12, 2019 18:19 PM

Suppose that we want to make the best Bayesian inference about some value $\mu$ we have some normal prior about it. I.e. $\mu\sim N(\mu_0, \mu_0)$ with known parameters. To do so, we can choose parameters $(\mu_x, \sigma_x)$ that define a normal sampling method with mean $\mu_s=\mu \frac{\sigma_x^2}{\sigma_x^2+\sigma_0^2}+\mu_x\frac{\sigma_0^2}{\sigma_x^2+\sigma_0^2}$ and variance $\sigma_s^2=\frac{\sigma_0^2\sigma_x^2}{\sigma_0^2+\sigma_x^2}$.

Can it be shown that the optimal value of $\mu_x$ is $\mu_x=\mu_0$?

Can it be shown that the optimal value of $\sigma_x$ is an intermediate value, i.e. $\sigma_x\in(0,\infty)$?

Can the optimal value of $\sigma_x$ be characterized in closed form?

Details: to be more precise, after choosing parameters $(\mu_x,\sigma_x)$ we will get an observation from the induced normal distribution and the goal is for the expected posterior to be as close as possible to $\mu$

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