Imagine I'm polling a random sample from the population and it asks them if they approve of the President or not. I also ask them some categorical demographic questions (age-bracket, race, gender, income-bracket).
Now given a new randomly-selected person from the population, I want to know the aposteriori probability distribution that he approves.
I take it that if this is all I know, the answer is just $Beta[approvers+1, nonapprovers+1]$ (assuming a uniform prior).
But I happen to know all the demographic information for this person too -- it's a 24-to-34-year-old white man in the lowest income bracket. Now I could just look at the 24-to-34-year-old low-income white men I polled, but I only polled one (or no) other person like that, leaving me essentially just with my prior. How do I appropriately combine all the information I have about different demographics and sub-demographics?