Instead of Bayesian updating of information, we propose to use the likelihood directly as a vehicle for coherent learning. Data concerning a one-dimensional interest parameter might be summarised in a likelihood function reduced of nuisance parameters. This reduced likelihood is combined with the likelihood of future data to update information. In the frequentist tradition, statistical reporting is often done in the format of confidence intervals.
Confidence distributions based on prior data represent frequentist analogues to Bayesian priors.
These confidence distributions need to be converted to likelihoods before they can be integrated with the new data likelihood. It is the statistical model, usually through a pivot, that dictates both the confidence distribution and the reduced likelihoods. Confidence distributions resulting from the integrated analysis, along with their probability bases, represent the frequentist analogue to the Bayesian posterior distributions. Asymptotics or bootstrapping is used to find pivots and their distributions, and hence reduced likelihoods and confidence distributions. A simple form of inverting bootstrap distributions to approximate pivots of the abc type is proposed.