4.3 Lindley’s method

4.3.1 A compromise with classical statistics

The following method appears first to have been suggested by Lindley (1965, Section 5.6), and has since been advocated by a few other authors, for example, Zellner (1971, Section 10.2; 1974, Section 3.7).

Suppose, as is common in classical statistics, that you wish to conduct a test of a point (or sharp) null hypothesis

Suppose further that your prior knowledge is vague or diffuse, so that you have no particular reason to believe that rather than that where is any value in the neighbourhood of .

The suggested procedure depends on finding the posterior distribution of θ using a reference prior. To conduct a significance test at level it is then suggested that you find a highest density region (HDR) from the posterior distribution and reject if and only if is outside this HDR.

4.3.2 Example

With the data on the uterine weight of rats which you met in Section 2.8 on ‘HDRs for the normal variance’, we found the posterior distribution of the variance to be

so that an interval corresponding to a 95% HDR for is (19, 67). Consequently, on the basis of the data, there you should reject a null hypothesis at the 5% level, but on the other hand, you should not reject a null hypothesis at that level.

4.3.3 Discussion

This procedure is appropriate only when prior information is vague or diffuse and even then it is not often the best way of summarizing posterior beliefs; clearly the significance level is a very incomplete expression of these beliefs. For many problems, including the one considered in the above example, I think that this method is to be seen as mainly of historical interest in that it gave a way of arriving at results related to those in classical statistics and thus helped to wean statisticians brought up on these methods towards the Bayesian approach as one which can get results like these as special cases, as well as having its own distinctive conclusions. However, it can have a use in situations where there are several unknown parameters and the complete posterior is difficult to describe or take in. Thus, when we come to consider the analysis of variance in Sections 6.5 and 6.6, we shall use the significance level as described in this section to give some idea of the size of the treatment effect.