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?

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.
