Cromwell’s rule (also known as the zero priors paradox) states that you should never assign prior probabilities of 0 or 1 to anything, no matter how likely or unlikely, unless it is a logically true or false statement. For example, 5 + 5 = 10 is a logically true statement, so can be assigned a probability of 1. “It’s going to rain this month” seems like a logically true statement (especially in the U.S. and England). But ask anyone who has been through a drought (like the English drought of 1976) and they will tell you to leave a little room for error.
What’s Behind the Name?
Cromwell’s rule was named by statistician Dennis Lindley after the statesman Oliver Cromwell.
While Cromwell never put down a mathematical formation of this rule, he is credited with a dramatic historical rendition of it. In August of 1650— the English executed Charles I and the Scots invited his son Charles II to become king of Scotland. Cromwell wrote a letter to the synod of the Church of Scotland asking them to reconsider their position. His words:
“I beseech you, in the bowels of Christ, think it possible that you may be mistaken.”
The Scots were convinced they were 100% certain that Charles II should be king. There was zero uncertainty, and Cromwell’s letter begged them to change from that “100% probability” stance that they were right.
Rationale Behind Cromwell’s Rule
By Bayes Rule, if a prior probability of 0 or 1 is assigned to any hypothesis, the posterior probability will necessarily be 0 or 1. No amount of data or evidence will be able to change it. This is obviously a danger in research science where conclusions should be based on research findings, not on prior beliefs.
Dennis Lindley’s often quoted way of phrasing the problem is this:
“…if a decisionmaker thinks something cannot be true and interprets this to mean it has zero probability, he will never be influenced by any data, which is surely absurd. So 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.”
Draper, David. Applied Mathematic and Statistics 206 Homework Set: Cromwell’s Rule
Retrieved from https://ams206-winter18-01.courses.soe.ucsc.edu/system/files/attachments/ams-206-quiz-3.pdf on February 27, 2018
Jackman, Simon. Bayesian Analysis for the Social Sciences.
Retrieved from https://books.google.com/books?id=QFqyrNL8yEkC on February 27, 2018
Lindley, Dennis (1991). Making Decisions (2 ed.). Wiley. p. 104. ISBN 0-471-90808-8.University of Houston, Empirical Implications of Theoretical Models. A (Very Brief) Introduction to Ideal Point Estimates
Retrieved from http://www.uh.edu/class/hobby/eitm/_docs/Point%20Estimates_EITM2016.pdf on February 27, 2018
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