Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 3 / e / Solution

The first coefficient is the posterior mean log odds of choosing invertebrates rather than fish for smaller alligators at Lake Hancock. It corresponds tofor those invertebrate-to-fish odds. This is not the absolute probability of choosing the second category, because the other categories also enter the normalization.

The size coefficient adds $-1.524$ to the invertebrate-to-fish log odds on changing to the larger class, holding lake fixed. Its common-across-lakes odds ratio isabout a 78 percent reduction in the relative odds. Exponentiating a posterior mean log odds is a geometric summary, not the arithmetic posterior mean of the odds.

There are $(K-1)p=4\cdot 5=20$ free category coefficients plus $I=8$ group intercepts, giving 28 free parameters in the fitted Poisson model. The effective parameter count in DIC, $26.8$, is close to 28 and slightly smaller, consistent with some regularization or incomplete information. Comparing it with 20 would omit the nuisance intercepts. A direct conditional multinomial fit has 20 coefficients but a different observational likelihood, since it conditions on the totals.