Imagine we are performing Bayesian inference with normal-normal conjugate priors. We have some prior:
$$ \mu \sim N(\mu_0, \sigma_0^2). $$
We know we will collect some normally distributed data $x$ with known variance $\sigma^2$ distributed as:
$$ x|\mu \sim N\left(\mu, \sigma^2\right). $$
The prior predictive distribution tells us the distribution of the observation $x$ prior to actually collecting the data:
$$ x \sim N\left(\mu_0, \sigma_0^2 + \sigma^2\right). $$
Once we actually collect a realization of $x$, we will use it to obtain the posterior distribution of $\mu$ using standard Bayesian updating:
$$ \mu~|~x \sim N\left(\frac{\mu_0/\sigma_0^2+x/\sigma^2}{1/\sigma_0^2+1/\sigma^2}, \frac{1}{1/\sigma_0^2+1/\sigma^2}\right). $$
Before actually collecting the data $x$, we know that our posterior variance is going to be $\frac{1}{1/\sigma_0^2+1/\sigma^2}$, but we are unsure what our posterior mean will be, since $x$ is a random variable prior to actually collecting the data. However, from the prior predictive distribution, we do know the distribution of the posterior mean before actually collecting the data $x$:
$$ \frac{\mu_0/\sigma_0^2+x/\sigma^2}{1/\sigma_0^2+1/\sigma^2} \sim N\left(\mu_0, \sigma_0^2 - \frac{1}{1/\sigma_0^2+1/\sigma^2}\right). $$
This leads to my terminology question. What is the name of the distribution of the posterior mean that you have before you actually collect your data? Clearly it's conceptually related to the prior predictive distribution, but I'm interested in the distribution of the posterior mean ($\frac{\mu_0/\sigma_0^2+x/\sigma^2}{1/\sigma_0^2+1/\sigma^2}$ in the example here) instead of the distribution of the observations themselves ($x$ in the example here).
For a bit of motivation of how this sort of question might come up, consider a clinical trial intermediate analysis. We have some data collected thus far, encoded in our prior $\mu\sim N(\mu_0, \sigma_0^2)$. We want to decide whether to run the trial another several months; if we do, we'll collect the additional observation $x$ and obtain some posterior mean. The distribution of the posterior mean before actually collecting the data can inform whether there's much chance of the trial crossing some decision boundary, which can inform whether to continue the trial (and collect $x$) or stop it for futility (and not collect $x$).