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 chi-squared, but in practice it is more convenient to use tables of the corresponding values of chi-squared, and such tables can be found in the Appendix. In fact, it does not make much difference whether we look for regions of highest density of the inverse chi-squared distribution or of the log chi-squared distribution, but insofar as there is a difference it seems preferable to base inferences on the log chi-squared distribution.

2.8.2 Example

When we considered the normal distribution with unknown mean but known variance, we had to admit that this was a situation which rarely occurred in real-life examples. This is even more true when it comes to the case where the mean is known and the variance unknown, and it should really be thought of principally as a building block towards the structure we shall erect to deal with the more realistic case where both mean and variance are unknown.

We shall, therefore, consider an example in which the mean was in fact unknown, but treat it as if the mean were known. The following numbers give the uterine weight (in mg) of 20 rats drawn at random from a large stock:

It is easily checked that n=20, , , so that and

In such a case, we do not know that the mean is 21.0 (or at least it is difficult to imagine circumstances in which we could have this information). However, we shall exemplify the methodology for the case where the mean is known by analyzing this data as if we knew that the mean were . If this were so, then we would be able to assert that

All the information we have about the variance is contained in this statement, but of course it is not necessarily easy to interpret from the point of view of someone inexperienced with the use of statistical methods (or even of someone who is but does not know about the inverse chi-squared distribution). Accordingly, it may be useful to give some idea of the distribution if we look for a HDR. From the tables in the Appendix, we see that the values of chi-squared corresponding to a 95% HDR for log chi-squared are 9.958 and 35.227, so that the interval for is from 664/35.227 to 664/9.958, that is is the interval (19, 67). (We note that it is foolish to quote too many significant figures in your conclusions, though it may be sensible to carry through extra significant figures in intermediate calculations.) It may be worth comparing this with the results from looking at HDRs for the inverse chi-squared distribution itself. From the tables in the Appendix A, 95% HDR for the inverse chi-squared distribution on 20 degrees of freedom lies between 0 025 and 0 094, so that the interval for is from to , that is is the interval (17, 62). It follows that the two methods do not give notably different answers.