The following calculations are based on Jeffreys’s (1961, p. 269) Theory of Probability and elaborates why he choose to use a symmetric proper weighting function on the test relevant parameter the population effect size .
First, to relate the observed -value to the population effect size within , Jeffreys rewrote the likelihood of in terms of the effect size and . To calculate the weighted likelihood of he then choose to set . By assigning the same weighting function to as was done for , we obtain:
The remaining task is to specify , the weighting function for the test-relevant parameter. Jeffreys proposed his weighting function based on desiderata obtained from hypothetical, extreme data.
Predictive matching: Symmetric
The first “extreme” case Jeffreys discusses is when ; this automatically yields regardless of the value of . Jeffreys noted that a single datum cannot provide support for , as any deviation of from zero can also be attributed to our lack of knowledge of . Hence, nothing is learned from only one observation and consequently the Bayes factor should equal 1 whenever .
To ensure that whenever , Jeffreys (1961, p. 269) entered , thus, and into Eq. 2 and noted that equals , if is taken to be symmetric around zero. The proof assumes that and uses the transformation , thus, and . Hence, by symmetry of we get
By swapping the order of integration (Fubini) then yields an integral in terms of that yields the normalisation constant of a normal distribution, that is,
Hence, for symmetric around zero the weighted likelihood of is then equal to that of the null model whenever and .