Bayesian statistics for a simple finite population This question has probably been asked before but I couldn't find it so here we go.
Let's assume we have a finite statistical population of $N$ members $x_1... x_N$.
Then for sure $\mu = N^{-1}\sum_{i=1}^Nx_i$ is a fixed parameter of this population. How does it make any sense to assume $\mu$ has a certain distribution i.e. $\mu$ is actually a random variable?
 A: This is no different from any other Bayesian model.
While $\mu$ is a fixed quantity, your state of knowledge about $\mu$ is limited (unless you have observed all $x_1,x_2,\dots,x_N$) and you represent your limited knowledge about $\mu$ (perhaps indirectly) by a probability model.  For example, you could represent your prior state of knowledge by saying that the $x_i$'s are independent $N(\eta,\tau^2)$ conditional on $\eta$ and $\tau^2$.  Specifying your prior also on $\eta$ and $\tau^2$ this would translate to a prior (or a posterior) also on the finite population mean $\mu=\frac1n\sum_{i=1}^N x_i$ if you have observed none of (or only some of) the $x_i$'s.
A: The apparently complete population you have might only be a sample from a larger, potentially infinite or hypothetical "hyper-population" that's of interest.
For example, one tacit assumption of your setup is that $x_i$ is exact. This is rarely the case. Even if you can measure to the gram, a person's body mass fluctuates by a few pounds over the course of a day. Test scores are even worse.
Similarly, the complete population might not actually exist. If you wanted to know whether American 4th graders were taller than Canadian 4th graders, you would probably sample a few classes from each country and do some inference. If you were mono-manically obsessed, you might try to do a complete census of the kids in each country, but that doesn't really work. There will be new 4th graders next year, and the year after that, and so on.
