  A theory of quantitative inference about the parameters of sampling
distributions is constructed deductively by following very general rules,
referred to as the Cox-Polya-Jaynes Desiderata. The inferences are made in
terms of probability distributions that are assigned to the parameters. The
Desiderata, focusing primarily on consistency of the plausible reasoning, lead
to unique assignments of these probabilities in the case of sampling
distributions that are invariant under Lie groups. In the scalar cases, e.g. in
the case of inferring a single location or scale parameter, the requirement for
logical consistency is equivalent to the requirement for calibration: the
consistent probability distributions are automatically also the ones with the
exact calibration and vice versa. This equivalence speaks in favour of
reconciliation between the Bayesian and Frequentist schools of reasoning.
