Terminology for distribution of posterior mean before seeing data

Question

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).

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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 $x$).

Answer 1

What is the name of the distribution of the posterior mean that you have before you actually collect your data?

The name is prior mean. If you don't have any data, the only thing you have are the priors, so there's only the prior distribution. You cannot update the prior if there is no data. There is no intermediate state.

I want to know how far the posterior mean could plausibly move away from the prior mean $\mu_0$ once I collect the data $x$ [...]

I'm afraid that there is no simple answer, it could go almost arbitrarily far. Recall that the posterior is the likelihood times the prior

$$ p(\theta | x) \propto p(x|\theta) \, p(\theta) $$

So the more informative, i.e. stronger the likelihood $p(\theta|x)$ is, the more it influences the result, the same applies to the prior $p(\theta)$, they both impact the result. Now, if your prior is flat $p(\theta) \propto 1$, it has no impact and the likelihood will dominate the result. If it is other "uninformative" prior it would have a limited impact. Now consider another, extreme case where the prior is a degenerate distribution that has all its probability mass centered at $\theta_0$ and is zero everywhere else. In such a case, the posterior could only be a degenerate distribution centered at $\theta_0$ regardless of the data. So the end result would depend on how informative your prior is, and how informative your data is.

If you want to answer the question for a specific model, the easiest approach would be a simulation study where you simulate different plausible values of $x$ and update the prior to see what are the possible posteriors.