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 ...
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