The GAMPL Procedure

Model Inference

Wahba (1983) proposes a Bayesian covariance matrix for parameter estimates $\bbeta $ by interpreting a smoothing spline as a posterior mean. Nychka (1988) shows that the derived Bayesian posterior confidence limits work well from frequentist viewpoints. The Bayesian posterior covariance matrix for the parameters is

\[ \bV _{\bbeta }=(\bX ’\bW \bX +\bS _{\blambda })^{-1}\sigma ^2 \]

The posterior distribution for $\bbeta $ is thus

\[ \bbeta |\mb{y} \sim N(\hat{\bbeta },\bV _{\bbeta }) \]

For a particular point whose design row is vector $\mb{x}$, the prediction is $\mb{x}\hat{\bbeta }$ and the standard error is $\sqrt {\mb{x}\bV _{\bbeta }\mb{x}’}$. The Bayesian posterior confidence limits are thus

\[ \left(\mb{x}\hat{\bbeta } \pm z_{\alpha /2} \sqrt {\mb{x}\bV _{\bbeta }\mb{x}’}\right) \]

where $z_{\alpha /2}$ is the $1-\alpha /2$ quantile of the standard normal distribution.

For the jth spline term, the prediction for the component is formed by $\mb{x}_ j\hat{\bbeta }$, where $\mb{x}_ j$ is a row vector of zeros except for columns that correspond to basis expansions of the jth spline term. And the standard error for the component is $\sqrt {\mb{x}_ j\bV _{\bbeta }\mb{x}’_ j}$.