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?

A saturated model is one in which there are as many estimated parameters as data points. By definition, this will lead to a perfect fit, but will be of little use statistically, as you have no data left to estimate variance.

For example, if you have 6 data points and fit a 5th-order polynomial to the data, you would have a saturated model (one parameter for each of the 5 powers of your independant variable plus one for the constant term).
