What is Cromwell's rule and why is it important for Bayesians?

Question

I have just heard of Cromwell's rule, but I'm not sure I understand it very well. What is Cromwell's rule and why is it important for Bayesian statistics?

Answer 1

As pointed out by others, this rule derives from the statement by the famous English General Oliver Cromwell, who stated: "I beseech you, in the bowels of Christ, think it possible that you may be mistaken."

Applied to a the context of Bayesian analysis, it recommends generally against assigning a prior probability of zero or one to an unknown event. In lieu of this, it is recommended that events thought to be highly implausible should be assigned an extremely low probability rather than zero probability. For a hypothesis $H$ and potential evidence $E$ Bayes' rule says that the (posterior) probability of the hypothesis given observation of that evidence is:

$$\mathbb{P}(H|E) = \frac{\mathbb{P}(E|H) \cdot \mathbb{P}(H)}{\mathbb{P}(E|H) \cdot \mathbb{P}(H) + \mathbb{P}(E|\bar{H}) \cdot \mathbb{P}(\bar{H})}.$$

If $\mathbb{P}(E|\bar{H})>0$ then using the prior probability $\mathbb{P}(H)=0$ gives the posterior probability $\mathbb{P}(H|E) = 0$, regardless of the likelihood ratio $\mathbb{P}(E|H)/\mathbb{P}(E|\bar{H})$. This is generally seen to be a drawback of this prior stipulation, because it means that you retain the view that the hypothesis has zero probability no matter how high the likelihood ratio gets.

Whilst the rule is a useful suggestion in some contexts, pretty much every Bayesian model restricts its attention to events considered possible within a particular model form and ascribes zero probability to all other events not considered in the analysis. Moreover, any analysis involving parameters that are treated as continuous variables will necessarily involve events with probability zero on these variables. Consequently, you cannot take the injunction seriously as a necessary requirement of analysis. In some instances, you may formulate a model in which a particular event is excluded from consideration and has zero probability (or a continuous outcome has zero probability density).

Here it is worth noting that Bayesian analysis has a "get out of jail free" card in cases where such an event is observed; if we observe an "impossible" event then any posterior belief is valid (see e.g., here). In the above case, this occurs in the special case where $\mathbb{P}(E|\bar{H})=0$. In this case the evidence $E$ is regarded as having zero probability unless the hypothesis is true. In this case a prior stipulation that $\mathbb{P}(H)=0$ is compatible with any posterior probability $\mathbb{P}(H|E)$ between zero and one.

Answer 2

According to Wikipedia, the rule states that:

the use of prior probabilities of 1 ("the event will definitely occur") or 0 ("the event will definitely not occur") should be avoided, except when applied to statements that are logically true or false, such as 2+2 equaling 4 or 5.

The motivation for the rule is that if you assign a prior probability of 0 to an event $A$, then no amount of evidence $B$ can ever increase that probability to anything greater than 0, because by Bayes' Theorem:

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0}{P(B)} = 0 $$

(You might object that $P(B)$ might also be 0, but that still doesn't provide a way to update your probability of $A$.)

An example: suppose $A$ is the event "man-sized ants will attack on Monday", and your prior probability for $A$ is $P(A) = 0$ because you think this is impossible. Suppose that on Monday you look out of the window and see a lot of man-sized ants attacking people. You still won't be able to assign $A$ a posterior probability greater than 0, so you won't be able to logically justify defending yourself from the ants.