Suppose that I culture cancer cells in n different dishes g₁, g₂, … , gn and observe the number of cells ni in each dish that look different than normal. The total number of cells in dish gi is ti. There is individual differences between individual cells, but also differences between the populations in different dishes because each dish has a slightly different temperature, amount of liquid, and so on.
I model this as a beta-binomial distribution: ni ~ Binomial(pi, ti) where pi ~ Beta(α, β). Given a number of observations of ni and ti, how can I estimate α and β?