Cromwell's rule
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Cromwell's rule, named by statistician Dennis Lindley, states that one should avoid using prior probabilities of 0 or 1, except when applied to statements that are logically true or false. (For instance, Lindley would allow us to say that 
)
The reference is to Oliver Cromwell, who famously wrote to the synod of the Church of Scotland on August 5, 1650 saying
- I beseech you, in the bowels of Christ, think it possible that you may be mistaken.
As Lindley puts it, if a coherent Bayesian attaches a prior probability of zero to the hypothesis that the Moon is made of green cheese, then even whole armies of astronauts coming back bearing green cheese cannot convince him. Setting the prior probability (what is known about a variable in the absence of some evidence) to 0 (or 1), then, by Bayes' theorem, the posterior probability (probability of the variable, given the evidence) is forced to be 0 (or 1) as well.
It is not inconceivable for something to have probability 0, but in the real world, virtually nothing does. However, many things do seem to have a probability of 1, which implies that the probability for these events not existing would be zero.
Cromwell's rule is necessary because otherwise the unique multiplication and division characteristics of zero would make the transformative effect of Bayes' Theorem nonexistent at a value of zero for prior probabilities. Arguably, redefining multiplication and division by zero could solve this problem, but at the price of a disarray of algebraic concepts as we know them.
[edit] References
- Lindley, D. Making Decisions. John Wiley. Second Edition 1985. ISBN 0-471-90808-8
