Interpretation of Bayesian " Confidence " Intervals for Smootidng Splines

A frequency interpretation of Bayesian "confidence" intervals for smoothing splines. Abstract The frequency properties of Wahba's Bayesian "confidence" intervals for smoothing splines are investigated by a large sample approximation and by a simulation study. When the coverage probabilities for these pointwise confidence intervals is averaged across the observation points, we explain why the average coverage probability (ACP) should be close to the nominal level. From a frequency point of view, the ACP is close to the nominal level because the average posterior variance for the spline is similar to a consistent estimate of the mean squared error and also because the mean squared bias is a modest fraction of the total mean squared error. These properties are independent of the Bayesian assumptions used to derive this confidence procedure and explain why the ACP is accurate for functions that are much smoother than the sample paths prescribed by the prior. The main disadvantage with this approach is that these confidence intervals are only valid in an average sense and may not be reliable if only evaluated at peaks or troughs in the estimate. This analysis accounts for the choice of the smoothing parameter (bandwidth) using cross validation and in the case of natural splines, we consider an adaptive method for avoiding boundary effects. SECTION 1 Consider the additive model: k=l,n where the observation vector Y =(Yl'YZ, • •. ,Y n) depends on a smooth, unknown function f evaluated at the points: 0 ~ t l ~ t z ... ~ t n ~ 1 and a vector of independent and identically distributed errors: I Z = Q and E(~ ~) = u I. The statistical problem posed by this model is to estimate f from y without having to to assume that f is contained in a specific parametric family. One solution is a smoothing spline estimate for f where the appropriate amount of smoothing is determined by generalized cross validation. Splines have been used successfully in a diverse range of applications and eventually may provide an alternative to standard parametric regression models. One limitation in applying spline methods in practice, however, is the difficulty in constructing confidence intervals or specifying other measures of the estimates accuracy. Wahba (1983), using the elegant interpretation of a smoothing spline as a posterior mean when f is viewed as the realization of a Gaussian stochastic process, suggested a pointwise confidence interval for f(t k) …