What are the assumptions for the Bayesian average? How does one derive the Bayesian average (https://en.wikipedia.org/wiki/Bayesian_average). As I understand it, this should derivable from the prior probability distribution for $\bar{x}$ and the likelihood function of a sample given a particular $\bar{x}$. However, I'm not able to figure out what prior and likelihood lead to formula in the link.
 A: The linked definition of "Bayesian Average" appears to be essentially heuristic,  and it is not entirely clear what you are asking.
However one possible interpretation is in the context of MAP estimation of the mean parameter $\mu$ for a Gaussian population. In this context the sample mean
$$\bar{x}=\frac{1}{N}\sum_ix_i$$
is a MLE for $\mu$.
To get the linked formula from a MAP framework, you could use a Gaussian prior for $\mu$ such that the MAP estimate becomes
$$\mu_\text{MAP}|\vec{x}=\frac{N\bar{x}+N_0\mu_0}{N+N_0}$$
The required prior is then
$$\mu\sim\text{N}(\mu_0,\sigma^2/N_0)$$
where $\sigma^2$ is the variance used in the likelihood, i.e. $x\sim\text{N}(\mu,\sigma^2)$.
So it is like specifying the "prior sample size" $N_0$ in terms of a "standard error".
(Note however that the formula is really heuristic, but that is OK, and it is reasonable without needing this sort of circuitous and assumption-laden justification!)
A: Your prior belief is that the average of a typical dataset with $C$ samples is equal to $m$. Your real dataset has $n$ samples $x_1,\ldots,x_n$. Therefore, combining your prior belief with this dataset you would have $C+n$ samples with a sum equal to $Cm+\sum_{i=1}^{n} x_i$. Therefore, the average of your dataset considering the prior belief would be equal to
$$\bar x=\frac{Cm+\sum_{i=1}^{n} x_i}{C+n}$$
