For Bayesian estimation of the binomial parameter, when the aim is to "let the data speak for themselves", the uniform or Bayes-Laplace prior appears preferable to the reference/Jeffreys prior recommended by objective Bayesians like Berger and Bernardo.

Here confidence intervals tend to be "exact" or "approximate", aiming for either minimum or mean coverage to be close to nominal. The latter criterion tends to be preferred, subject to "reasonable" minimum coverage. I will first re-iterate examples of how the highest posterior density (HPD) credible interval based on the uniform prior appears to outperform both common approximate intervals and Jeffreys prior based intervals, which usually represent credible intervals in review articles.

Second, an important aspect of the recommended interval is that it may be seen to be invariant under transformation when taking into account the likelihood function. I will also show, however, that this use of the likelihood does not always lead to excellent, or even adequate, frequentist coverage.

Third, this approach may be extended to nuisance parameter cases by considering an "appropriate" likelihood of the parameter of interest. For example, quantities arising from the 2x2 contingency table (e.g. odds ratio and relative risk) are important practical applications, apparently leading to intervals with better frequentist performance than that found for HPD or central credible intervals. Preliminary results suggest the same for "difficult" problems such as the ratio of two Normal means ("Fieller-Creasy") and the binomial N problem.