Head of the Methodology and Statistics Group,
Competence Center for Methodology and Statistics (CCMS), Luxembourg
This paradox is clearly inspired by and in a sense is just another form of Philip Dawid’s selection paradox. See my paper in The American Statistician for a discussion of this. However, I rather like this concrete example of it.
Imagine that you are about to carry out a Bayesian analysis of a new treatment for rheumatism. However, just to avoid various complications I am going to assume that you are looking at a potential side effect of the treatment. I am going to take the effect on diastolic blood pressure (DBP) as the example of a side-effect one might look at.
Now, to be truly Bayesian I think that you ought to have a look at a long list of previous treatments for rheumatism but time is short and this is not always so easy. So instead you argue like this.
- I know from the results of the WHO Monica project that the standard deviation of DBP is about 11mmHg in a general population.
- I have no prior opinion as to whether anti-rheumatics as a class have a beneficial or harmful effect on DBP
- I think that large effects on DBP, whether harmful or beneficial, are rather improbable for a drug designed to treat rheumatism.
- I believe the data are approximately Normal
- I am going to use a conjugate prior for the effect of treatment with mean 0 and standard deviation = 4 mm Hg. This makes very large beneficial or harmful effects unlikely but still allows reasonable play for the data. This means that the prior variance is 16mgHg2 compared to a data variance I am expecting to be about 120 mmHg2. This means that as soon as I have treated 8 subjects the data mean variance should be smaller (about 15 mmHg2) that the prior mean and so I will actually be weighting the data more than the prior at that point. This seems about reasonable to me.
You can choose different figures if you want but here I am attempting to apply a standard Bayesian analysis in a reasonably honest manner. Continue reading