Home > Regression > Modeling Field Goal Kicking — part 2

Modeling Field Goal Kicking — part 2

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 \beta_0 and \beta_1 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
g(p(30), p(50)) = beta(p(30), a_1, b_1) beta(p(50), a_2, b_2))
where a_1, b_1, a_2, b_2 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 (\beta_0, \beta_1).  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.
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Categories: Regression
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