Consider the following Bayesian model

with prior for F as

where , and are positive constants, and is a zero mean Gaussian stochastic process independent of with covariance . Wahba (1978) showed that with . Formulae for computing posterior means and variances were provided in Gu and Wahba (1993b). Posterior variances can be used to construct confidence intervals for :

where is the quantile of a standard normal distribution (Wahba, 1990). The intervals defined in () are referred to as the Bayesian confidence intervals (Wahba, 1983). These Bayesian confidence intervals are not point-wise confidence intervals. Rather, they provide across-the-function coverage (Nychka, 1988; Wang and Wahba, 1995).

Often one needs to test

This hypothesis is equivalent to or . Three tests were considered in Wahba (1990): locally most powerful (LMP), GCV and GML tests. Let

be the QR decomposition of , and be the eigenvalue decomposition of with eigenvalues . Let . Then the test statistics for LMP, GML and GCV tests are

and

where . It can be shown that under the corresponding Bayesian model, the LMP test is the score test and the GML test is the likelihood ratio test. Furthermore, the GCV test is closely related to the F-test based on the extra sum of squares principle (Liu and Wang, 2004). Usually the p-values cannot be calculated analytically because the null distributions under are unknown. Standard theory for likelihood ratio tests does not apply because the parameter is on the boundary under the null hypothesis. The non-standard asymptotic theory developed by Self and Liang (1987) does not apply either because of the lack of replicated observations. Monte Carlo method can be used to approximate p-values. However, they are usually computational intensive since the smoothing parameter needs to be estimated for each Monte Carlo sample. In the current version, through the utility function

An alternative approach to visually check above hypothesis is to plot
the projection of onto together with its Bayesian
confidence intervals. When is true, most parts of the zero function
should be inside these confidence intervals. See Section 7 for examples.
Two utility functions, `predict.ssr` and `plot.bCI`, are available to
compute posterior means, standard deviations and plot fits with Bayesian
confidence intervals. See help files of `predict.ssr` and `plot.bCI`
for more details.