(Villegas (1990)) Consider a family of probability distributions   with  taking values in a Euclidean affine space E, such that the likelihood function is

This model is called Euclidean Bayesian if 

a. Deduce that the corresponding Euclidean prior distribution in the case of a Poisson model

b. Show that the p-value    when testing   against  is related to this prior distribution, but that this relation does not hold for the alternative test of

 against 

c. Show that the Haldane prior distribution

also appears as a Euclidean model when   still the Euclidean prior for the negative binomial distribution, N eg(n, p)?

d. If   show that, in the binomial case, the p-values   and   associated with the hypotheses   do not correspond to the Euclidean distribution (9.7.3).

e. In the normal case   show that the Euclidean prior distributions are the following ones:

(i) 

(ii)   and

(iii) 

Examine the issue of compatibility between the invariance requirements and the Likelihood Principle. In particular, determine whether the maximum likelihood estimator is always an invariant estimator.