By Peter J. Green, Bernard W. Silverman (auth.)

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40) above, and so the spline smoother g is the posterior mode given the data. Wahba (1978, 1983), drawing on earlier work ofKimeldorf and Wahba (1970), developed this approach in regression, and suggested the use of pointwise error bands for the curve estimate based on the posterior distribution. We now summarize some results of these papers in the next section. A Gaussian process prior Both the prior and the posterior log densities are quadratic forms in the function g, and so they correspond to a Gaussian process structure.

Tn < b. Suppose that the smoothing parameter a and the weights w;, i = 1, ... , n are all strictly positive. Given data values Yt, ... , Yn. the penalized weighted sum of squares Sw(g) is uniquely minimized over gin Sl[a, b] by the natural cubic spline with knots at the points t; having g = (W + aK)- 1WY. 2, replacing the residual sum of squares by the weighted residual sum of squares. 22). The details are left as an exercise for the reader. 22) directly for calculation, and in the next section we set out the extension of the Reinsch algorithm to incorporate weights.

2. QT)g = Y. QT g; now substitute QT g =R-y and simplify to give an explicit formula for g in terms of Y and '"Y g = y- aQ-y. 14) Again using the condition QT g =R-y we obtain QTY- aQT Q-y =R-y, which gives the equation for '"Y (R + aQT Q)'"Y = QTY. 15) This equation is the core of the algorithm. 13) for g, it can be solved in linear time using band matrix techniques. The matrix (R + aQT Q) is easily seen to have bandwidth 5, and also to be symmetric and strictly positive-definite. Therefore it has a Cholesky decomposition of the form R+aQTQ=WLT where D is a strictly positive diagonal matrix and L is a lower triangular band matrix with Lij = 0 for j < i- 2 andj > i, and Lu = 1 for all i.