Wahba (1983) proposes a Bayesian covariance matrix for parameter estimates 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
The posterior distribution for is thus
For a particular point whose design row is vector , the prediction is and the standard error is . The Bayesian posterior confidence limits are thus
where is the quantile of the standard normal distribution.
For the jth spline term, the prediction for the component is formed by , where 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 .