I got stuck with this problem,
Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$, where $\sigma^2$ is known and $\mu$ is unknown and will be updated using Bayesian inference.
We give a prior distribution of $\mu \sim \mathcal{N}(\mu_0, \sigma_0^2)$ and update it after observation $x_0$ to
\mu \sim \mathcal{N}(\mu'_0, \sigma_0^2), \;\;\; \mu'_0 = \frac{\sigma_0^2 x_0 + \sigma^2 \mu_0}{\sigma_0^2 + \sigma^2}
The problem is, actually we don't have any observations but want to see the result over all possible situations,
$U = \int p(x_0) \mathbb{E}(X|x_0) dx_0$
which is used in the Dynamic programming formulation for the forward planning.
I want to ask,
1- does this $U$ equation even make sense?
2- what's the final analytical form of $U$, if it makes sense? I can expand it as,
$U = \int \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \{ - \frac{(x_0 - \mu)^2}{2 \sigma^2} \} \frac{\sigma_0^2 x_0 + \sigma^2 \mu_0}{\sigma_0^2 + \sigma^2} dx_0 $
but don't have a clue to how to proceed.
Thanks for your help!