  Cromwell's rule (also known as the zero priors paradox) refers to the
constraint of classical probability theory that if one assigns a prior
probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as
well (this is a straightforward implication of how Bayes's rule works).
Relatedly, hypotheses with a very low prior cannot be updated to have a very
high posterior without a tremendous amount of new evidence to support them (or
to make other possibilities highly improbable). Cromwell's rule appears at odds
with our intuition of how humans update probabilities. In this work, we report
two simple decision making experiments, which seem to be inconsistent with
Cromwell's rule. Quantum probability theory, the rules for how to assign
probabilities from the mathematical formalism of quantum mechanics, provides an
alternative framework for probabilistic inference. An advantage of quantum
probability theory is that it is not subject to Cromwell's rule and it can
accommodate changes from zero or very small priors to significant posteriors.
We outline a model of decision making, based on quantum theory, which can
accommodate the changes from priors to posteriors, observed in our experiments.
