Let’s continue the football kicking example.
1. Prior beliefs. On the surface, it seems challenging to talk about prior information since the regression parameters and are not very meaningful. But we might have some beliefs about the probability of kicking a goal successful and we can use these beliefs to indirectly construct a prior on the regression coefficients.
2. Conditional means prior. We instead consider the parameters (p(30) and p(50)), the probabilities of successfully kicking a field goal at 30 yards and 50 yards. After some thought, I believe:
- The median and 90th percentile at p(30) are respectively 0.90 and 0.98 (I’m pretty confident of a successful kick at 30 yards.)
- The median and 90th percentile at p(50) are respectively 0.50 and 0.70. (I’m less confident of a successful kick of 50 yards.)
Assuming my beliefs about p(30) and p(50) are independent, and assuming beta priors, my joint prior is given by
are the matching beta shape parameters found using a function like beta.select in the LearnBayes package.
3. Then we can transform this prior on (p(30), p(50)) to a density on
. It can be shown that this transformed prior has a product of betas form which makes it very convenient to use.
The log of the logistic likelihood is actually a function logisticpost in the LearnBayes package, and it is convenient to use it for this example.