Say I have two coins from a particular mint in the US.
I flip coin one 20 times and receive 4 heads, giving me a beta distribution for the bias of coin one of $Beta$($\alpha$=5, $\beta$=17).
I then flip coin two 200 times and receive 105 heads, giving me a beta distribution for coin two of $Beta$($\alpha$=106, $\beta$=96).
What I would like to calculate is (assuming a prior of $Beta$(1,1)) the probability distribution that represents the average bias of coins from that mint after flipping just the two coins (or generalizing to N coins w/ different beta distributions). There is no assumption that the coins have the same bias.
I want to know if/how this can be calculated analytically.
In case it helps, this is what I've used to answer the question via hierarchical model MCMC simulation (where mu is the parameter of interest):
Hyperpriors: k ~ $Gamma$(s=1, r= .1), mu ~ $Beta$(1, 1)
Prior: coin_bias$_i$ ~ $Beta$(mu*k,(1-mu)*k)
Likelihood: $y_i$ ~ $Binomial$(flips$_i$, heads$_i$, coin_bias$_i$)