The Art of Statistics: Learning from Data

by David Spiegelhalter

Robert Kass and Adrian Raftery are two renowned Bayesian statisticians who proposed a widely used scale for Bayes factors, shown in Table 11.3. Note the contrast to the scale in Table 11.2 for verbally interpreting likelihood ratios for legal cases, where a likelihood ratio of 10,000 was required to declare the evidence as ‘very strong’, in contrast to scientific hypotheses only needing a Bayes factor of greater than 150. This perhaps reflects the need to establish criminal guilt ‘beyond reasonable doubt’, whereas scientific claims are made on weaker evidence, with many being overturned on further research. Our chapter on hypothesis testing contained a claim that a P-value of 0.05 was only equivalent to ‘weak evidence’. The reasoning for this is partly based on Bayes factors: P = 0.05 can be shown to correspond, under some reasonable priors under the alternative hypothesis, to Bayes factors between 2.4 and 3.4, which Table 11.3 suggests is weak evidence. As we saw in Chapter 10, this led to a proposed reduction in the necessary P-value for claiming a ‘discovery’ to 0.005. Unlike null-hypothesis significance testing, Bayes factors treat the two hypotheses symmetrically, and so can actively support a null hypothesis. And if we are willing to put prior probabilities on hypotheses, we might even calculate posterior probabilities of alternative theories for how the world works. Suppose, based on theoretical grounds alone, we judged it 50:50 whether the Higgs boson existed, corr...

Bayes factor vs P value