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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