Posts Tagged With: Jack Good

Stephen Senn: A Paradox of Prior Probabilities

Stephen Senn

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[1]. See my paper in The American Statistician for a discussion of this[2]. 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.

  1. I know from the results of the WHO Monica project that the standard deviation of DBP is about 11mmHg in a general population.
  2. I have no prior opinion as to whether anti-rheumatics as a class have a beneficial or harmful effect on DBP
  3. I think that large effects on DBP, whether harmful or beneficial, are rather improbable for a drug designed to treat rheumatism.
  4. I believe the data are approximately Normal
  5. 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

Categories: Statistics | Tags: , , , | 13 Comments

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