Bayesian Statistics: An Introduction, 4th Edition

2.4 Dominant likelihoods

2.4.1 Improper priors

We recall from the previous section that, when we have several normal observations with a normal prior and the variances are known, the posterior for the mean is

where and are given by the appropriate formulae and that this approaches the standardized likelihood

insofar as is large compared with , although this result is only approximate unless is infinite. However, this would mean a prior density which, whatever θ0 were, would have to be uniform over the whole real line, and clearly could not be represented by any proper density function. It is basic to the concept of a probability density that it integrates to 1 so, for example,

cannot possibly ...