  Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370--418; 54 296--325]
introduced the observed likelihood function to statistical inference and
provided a weight function to calibrate the parameter; he also introduced a
confidence distribution on the parameter space but did not provide present
justifications. Of course the names likelihood and confidence did not appear
until much later: Fisher [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng.
Sci. 222 (1922) 309--368] for likelihood and Neyman [Philos. Trans. R. Soc.
Lond. Ser. A Math. Phys. Eng. Sci. 237 (1937) 333--380] for confidence. Lindley
[J. Roy. Statist. Soc. Ser. B 20 (1958) 102--107] showed that the Bayes and the
confidence results were different when the model was not location. This paper
examines the occurrence of true statements from the Bayes approach and from the
confidence approach, and shows that the proportion of true statements in the
Bayes case depends critically on the presence of linearity in the model; and
with departure from this linearity the Bayes approach can be a poor
approximation and be seriously misleading. Bayesian integration of weighted
likelihood thus provides a first-order linear approximation to confidence, but
without linearity can give substantially incorrect results.
