  In statistical practice, whether a Bayesian or frequentist approach is used
in inference depends not only on the availability of prior information but also
on the attitude taken toward partial prior information, with frequentists
tending to be more cautious than Bayesians. The proposed framework defines that
attitude in terms of a specified amount of caution, thereby enabling data
analysis at the level of caution desired and on the basis of any prior
information. The caution parameter represents the attitude toward partial prior
information in much the same way as a loss function represents the attitude
toward risk. When there is very little prior information and nonzero caution,
the resulting inferences correspond to those of the candidate confidence
intervals and p-values that are most similar to the credible intervals and
hypothesis probabilities of the specified Bayesian posterior. On the other
hand, in the presence of a known physical distribution of the parameter,
inferences are based only on the corresponding physical posterior. In those
extremes of either negligible prior information or complete prior information,
inferences do not depend on the degree of caution. Partial prior information
between those two extremes leads to intermediate inferences that are more
frequentistic to the extent that the caution is high and more Bayesian to the
extent that the caution is low.
