Instead of asking ‘what do my data show?’, the Bayesian analyst asks ‘how do my data alter our view of the world?'. It may not sound like much, but it is a fundamental change of outlook. The idea is that the results of the new study are assessed in the light of the existing data, to establish an updated assessment of parameter values and their uncertainties. There are now two models rather than one. There is a model for what we know already: this is called the prior. Then there is a model that we fit to our data: this is the likelihood. The two models combine to give us an estimate of the posterior. We use the posterior distribution to make statistical inferences. Under its Bayesian interpretation, probability measures our confidence that something is true.
When the new study is small, and existing knowledge is extensive, then we should not expect our work to make much difference to overall understanding. For instance, if we studied 150 subjects and found that age at death was not affected by their smoking habits, we would be ill advised to conclude that this result had great generality. This is because we know from studies of many thousands of subjects over many decades that smoking leads to a reduction of about 14 years in mean age at death. Small studies can be highly informative, of course, especially when they are original and address important questions. But the larger and better designed the new study, the more that study should be capable ...