Sunday, September 23, 2007
Conditional means prior
In an earlier post, we illustrated Bayesian fitting of a logistic model using a noninformative prior. Suppose instead that we have subjective beliefs about the regression vector. A convenient way of representing these beliefs is by use of a conditional means prior. We illustrate this for our math placement example.
First, we consider the probability p1 that a student in placement level 1 receives an A in the class. Our best guess at p1 is 0.05 and this belief is worth 200 observations -- we match this info to a beta(200*0.05, 200*0.95) prior. Next, we consider the probability p5 that a student in level 5 gets an A -- our guess at this probability is 0.15 and this guess is worth 200 observations. We match this belief to a beta(200*0.15, 200*0.85) prior. Assuming that our beliefs about p1 and p5 are independent, the joint prior on (p1, p5) is a product of beta densities. Transforming back to the (beta0, beta1) scale, one can show that the prior on beta is given by
g(beta) = p1^a1 (1-p1)^b1 p5^a2 (1-p5)^b2
(Note that the conditional means prior translates to the same functional form as the likelihood where the beta parameters a1, b1, a2, b2 play the role of "prior data".)
The below figure displays (in red) a sample from the likelihood and (in blue) a sample from the prior. Here we see some conflict between the prior beliefs and the data information about the parameter.