Bayes (1763) 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 restricted attention to models now called location models; of course the names likelihood and confidence did not appear until much later: Fisher (1922) for likelihood and Neyman (1937) for confidence. Lindley (1958) 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 seriously misleading. Bayesian integration of weighted likelihood provides a first order linear approximation to confidence, but without linearity can give substantially incorrect results. 1

The original Bayes proposal leads to likelihood and confidence for many simple examples. More generally it gives approximate confidence but to achieve exact confidence reliability it needs refinement of the argument and needs more than just the usual minimum of the likelihood function from observed data. A general

summary Parameter curvature was introduced by Efron (1975) for classifying curved exponential models. We develop an alternative definition that describes curvature relative to location models. This modified curvature calibrates how Bayes posterior probability differs from familiar frequency based probability. And it provides a basis for then correcting Bayes probabilities to agree with the reproducibility traditional to mainstream statistics. The two curvatures are compared and examples are given. Bayes calibration; Bayes-frequentist discrepancy; Efron curvature; Exponential model approximation; Location model approximation.

summary Parameter curvature was introduced by Efron (1975) for classifying curved exponential models. We develop an alternative definition that describes curvature relative to location models. This modified curvature calibrates how Bayes posterior probability differs from familiar frequency based probability. And it provides a basis for then correcting Bayes probabilities to agree with the reproducibility traditional to mainstream statistics. The two curvatures are compared and examples are given. Bayes calibration; Bayes-frequentist discrepancy; Efron curvature; Exponential model approximation; Location model approximation.

Bayesian and frequentist methodologies when applied to the same model–data information can lead to different statistical inference results. A prominent example involves a rotationally symmetric normal error distribution located at an arbitrary point (θ1,θ2) on the plane. The radial distance ρ = (θ 2 1 + θ2 2)1/2 from the origin has a Bayes posterior survival value s(ρ) that is uniformly greater than the frequentist p-value p(ρ), can be expressed in terms of the noncentral chi-square distribution function with 2 degrees of freedom, and can attain 8 percentage points when ˆρ = 5. We use this Bayes–frequentist difference as a reference to explore the Bayesian bias attributable to parameter curvature. For this, we consider a two parameter regular statistical model and define a curvature measure for an interest parameter; the curvature measure is a mofication of the Efron measure and targets Bayesian adjustment rather than departure from the information lower bound as considered by Efron. Examples are given and simulations are provided.

This note concerns the use of parametric bootstrap sampling to carry out Bayesian inference calculations. This is only possible in a subset of those problems amenable to MCMC analysis, but when feasible the bootstrap approach offers both computational and theoretical advantages. The discussion here is in terms of a simple example, with no attempt at a general analysis.