4.6 The Doogian philosophy
4.6.1 Description of the method
Good (1983, Chapter 4 and elsewhere) has argued in favour of a compromise between Bayesian and non-Bayesian approaches to hypothesis testing. His technique can be summarized as follows (in his own words):
The Bayes/non-Bayes synthesis is the following technique for synthesizing subjective and objective techniques in statistics. (i) We use the neo-Bayes/Laplace philosophy (i.e. the techniques described in Section 4.4 on point null hypotheses with prior information) in order to arrive at a factor F (which is 1/B in the notation used here) in favour of the non-null hypothesis. For the particular case of discrimination between two simple statistical hypotheses, the factor in favour is equal to the likelihood ratio [as was shown in Section 4.1 when hypothesis testing was first considered], but not in general. The neo-Bayes/Laplace philosophy usually works with inequalities between probabilities, but for definiteness we here assume that the initial distributions are taken as precise, though not necessarily uniform. (ii) We then use F as a statistic and try to obtain its distribution on the null hypothesis, and work out its tail area, P. (iii) Finally, we look to see if F lies in the range (1/30P, 3/10P). If it does not lie in this range we think again. (Note that F is here the factor against H.)
This is certainly not unworkable. For example, in Section 4.5 we found that
so that B is a monotonic function of z2, and hence the ...