Bayesian Statistics: An Introduction, 4th Edition

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 ...