2.8 HDRs for the normal variance
2.8.1 What distribution should we be considering?
It might be thought that as the normal variance has (under the assumptions we are making) a distribution which is a multiple of the inverse chi-squared distribution we should be using tables of HDRs for the inverse chi-squared distribution to give intervals in which most of the posterior distribution lies. This procedure is, indeed, recommended by, for example, Novick and Jackson (1974, Section 7.3) and Schmitt (1969, Section 6.3). However, there is another procedure which seems to be marginally preferable.
The point is that we chose a reference prior which was uniform in 
so that the density of 
was constant and no value of 
was more likely than any other a priori. Because of this, it seems natural to use 
in the posterior distribution and thus to look for an interval inside which the posterior density of log 
is higher than anywhere outside. It might seem that this implies the use of tables of HDRs of log ...
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