(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
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:
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.