Cromwell’s Rule: Simple Definition

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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 on February 27, 2018
Jackman, Simon. Bayesian Analysis for the Social Sciences.
Retrieved from 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 on February 27, 2018

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