D Some predictive and conditional pdfs
Pages 4

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


) p 2 √ |X ′X|

× Γ(a+ p 2 )


( 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.