# How do I calculate the aposteriori probability distribution for someone's answer to a poll being an approval?

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?

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@Michael: I agree there was something wrong with the question, since the assumptions behind the "probability distribution" hadn't been explicated, but simply replacing that by "probability" has made things worse -- now the beta distribution makes no sense. –  joriki Sep 12 '12 at 17:09
Okay @joriki, I see I need to change it to posterior probability distribution for the outcome. –  Michael Chernick Sep 12 '12 at 17:25
@Michael: I think it should be "the aposteriori probability distribution for the probability that he approves", no? There are some questionable implicit assumptions behind this, but at least it makes sense, whereas I don't understand what "the aposteriori probability distribution that he approves" means. –  joriki Sep 12 '12 at 17:40
@joriki Yes that should be in the title also. –  Michael Chernick Sep 12 '12 at 18:25