Empirical probabilities are not exactly 0 or 1

Cromwell’s Rule in statistics argues that no empirical proposition should be assigned a subjective probability of exactly \(0\) or \(1\) - it is always possible to be mistaken. (Some argue that this rule should be generalized to logical facts as well.)

A probability of exactly \(0\) or \(1\) corresponds to infinite log odds, and would require infinitely strong evidence to reach starting from any finite prior. To put it another way, if you don’t start out infinitely certain of a fact before making any observations (before you were born), you won’t reach infinite certainty after any finite number of observations involving finite probabilities.

All sensible universal priors seem so far to have the property that they never assign probability exactly \(0\) or \(1\) to any predicted future observation, since their hypothesis space is always broad enough to include an imaginable state of affairs in which the future is different from the past.

If you did assign a probability of exactly \(0\) or \(1,\) you would be unable to update no matter how much contrary evidence you observed. Prior odds of 0 : 1 (or 1 : 0), times any finite likelihood ratio, end up yielding 0 : 1 (or 1 : 0).

As Rafal Smigrodski put it:

“I am not totally sure I have to be always unsure. Maybe I could be legitimately sure about something. But once I assign a probability of 1 to a proposition, I can never undo it. No matter what I see or learn, I have to reject everything that disagrees with the axiom. I don’t like the idea of not being able to change my mind, ever.”


  • Bayesian reasoning

    A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.