Abstract:
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.