Subjective probability is a measure of the expectation that an event will occur, or that a statement is true. Probabilities are given a value between 0 (the event will definitely not occur) and 1 (the event is absolutely certain to occur). The nearer the probability of an event tends towards 1, the more certain it is that the event will occur. The nearer the probability tends towards 0, the more certain it is that the event will not occur.
Cromwell's rule, named by statistician Dennis Lindley, states that the use of prior probabilities of 0 or 1 should be avoided, except when applied to statements that are logically true or false. For instance, Lindley would allow us to say that Pr(2+2 = 4) = 1, where Pr represents the probability. In other words, arithmetically, the number 2 added to the number 2 will certainly equal 4.
- I beseech you, in the bowels of Christ, think it possible that you may be mistaken.
As Lindley puts it, assigning a probability should "leave a little probability for the moon being made of green cheese; it can be as small as 1 in a million, but have it there since otherwise an army of astronauts returning with samples of the said cheese will leave you unmoved." Similarly, in assessing the likelihood that tossing a coin will result in either a head or a tail facing upwards, there is a possibility, albeit remote, that the coin will land on its edge and remain in that position.
If the prior probability assigned to a hypothesis is 0 or 1, then, by Bayes' theorem, the posterior probability (probability of the hypothesis, given the evidence) is forced to be 0 or 1 as well; no evidence, no matter how strong, could have any influence.
Cromwell's Rule: Bayesian Divergence (pessimistic)
An example of Bayesian Divergence of opinion is in Appendix A of Sharon Bertsch McGrayne's 2011 book. The Theory That Would Not Die: How Bayes' Rule Cracked The Enigma Code, Hunted Down Russian Submarines, & Emerged Triumphant from Two Centuries of Controversy.  In McGrayne's example (suggested by Albert Mandansky), Tim and Susan disagree over whether a stranger (who has two fair coins and one unfair coin with two heads) is tossing one of the two fair coins or the unfair coin. The stranger has tossed the coin 3 times and it comes up heads each time. Tim's prior was that the stranger would pick the coin randomly with a 1/3 probability of picking the unfair coin. Applying Bayesian inference Tim calculates there is an 80% probability of the unfair coin. Susan has a problematic prior of either zero or one. Using Susan's zero or one prior the probability remains the same after the evidence. Tim and Susan's probabilities do not converge.
Cromwell's Rule: Bayesian Convergence (optimistic)
An example of Bayesian Convergence of opinion is in Nate Silver's 2012 book, The Signal and the Noise: Why so many predictions fail — but some don't. after stating, "Absolutely nothing useful is realized when one person who holds that there is a 0 [zero) percent probability of something argues against another person who holds that the probability is 100 percent," Silver describes a simulation where three investors start out with initial guesses of 10%, 50% and 90% that the stock market is in a bull market, by the end of the simulation (shown in a graph), "all of the investors conclude they are in a bull market with almost (although not exactly of course) 100 percent certainty."
- Jackman, Simon (2009) Bayesian Analysis for the Social Sciences, Wiley. ISBN 978-0-470-01154-6 (ebook ISBN 978-0-470-68663-8)
- Carlyle, Thomas, ed. (1855). Oliver Cromwell's Letters and Speeches 1. New York: Harper. p. 448.
- Lindley, Dennis (1991). Making Decisions (2 ed.). Wiley. p. 104. ISBN 0-471-90808-8.
- McGrayne, Sharon Bertsch. (2011). The Theory That Would Not Die: How Bayes' Rule Cracked The Enigma Code, Hunted Down Russian Submarines, & Emerged Triumphant from Two Centuries of Controversy. New Haven: Yale University Press. 13-ISBN 9780300169690/10-ISBN 0300169698; OCLC 670481486 The Theory That Would Not Die, pages 263-265 at Google Books
- McGrayne, Sharon Bertsch. "Bayes Examples (2nd example)". same as Appendix A. Retrieved 4/10/2013.
- Silver, Nate (2012). The Signal and the Noise: Why so many predictions fail -- but some don't. New York: Penguin. pp. 258–261. ISBN 978-1-59-420411-1.