ABSTRACT
Let’s denote the linear form in the log scale for the expected abundance λi
λi = Log(Xiβ)
Let the data y be distributed following a Poisson distribution
[y|β] = n∏ i
dPois(yi, λi)
and let the parameters β be a priori distributed following a multidimensional Student distribution
β = dmStudent( √ a(β − β0), (X ′X)−1, 2a)× a
Then the joint distribution [y, β] writes
[y, β] =
( n∏ i
e−λiλyii Γ(1 + yi)
)( 1
2pic
) p 2 √ |X ′X|
× Γ(a+ p 2 )
Γ(a)
( 1 +
(β − β0)′(X ′X)(β − β0) 2a
We detail here how the data augmentation approach of Albert and Chib [3] renders easy the inference of the ordered multinomial probit model presented on page 187 of Chapter 8 for the skate data.
