Here is a simple prediction problem. Suppose we observe the number of home runs and the number of at-bats for 20 baseball players during the first month of the baseball season (April).
We observe corresponding home run rates: Utley 4/88, Weeks 1/77, Aurila 4/71, and so on.
From this data, we want to predict the home run rates for the same 20 players for the next month (May).
How can we best do this? Here are some opening comments.
1. One idea would be to estimate the May rates just by using the April rates. So we predict Utley's May rate to be 4/88, Weeks rate to be 1/77, etc. This may not be a good idea since we have limited data for each player.
2. Or maybe a better strategy would be to combine the data. Collectively in April, this group hit 60 home runs in 1697 at-bats for a rate of 60/1697 = 0.035. We could estimate each player's May rate by 0.035. But this would ignore the fact that players have different abilities to hit home runs.
3. Actually, the best prediction strategy is a compromise between the first two ideas. A good plan is to predict a player's May rate by a "shrinkage" estimate that shrinks a player's individual rate towards the combined rate.
In the following postings, we'll illustate how to fit a Bayesian exchangeable model to these data that gives a good prediction strategy.