Let's say, for the sake of ease, I have a sample of data that I believe comes from the binomial distribution. Let's say I have 100 observations in my sample. Previous to this experiment, I have created 10 samples of data and recorded summary statistics on them (each sample having 10 observations). I want to use these old samples to form my prior beliefs. The number of trials for each observation is 10.
When I work out the mean of my 10 previous samples, I find it is about equal to 4.82. When I work out the average variance I get about 2.7.
I would like to use a beta distribution as my prior, for the sake of conjugacy. I set my prior mean and variance equal to the mean and variance of my samples (the average value that is). Then I use the formula that assumednormal posted here.
Now, I end up with a negative $\alpha$ value, which of course is impossible. Obviously, the mean of the beta distribution should be between 0 and 1, and this is violated due to my calculating of the mean of my samples. Where have I gone wrong and what is the correct remedy? (I think my mean should be 0.482, but I can't work out the justification to get here).
