Smoothing Splines and the Kalman Smoother. Consider the discrete time version of the smoothing spline argument given in (2.56); that is, suppose we observe yt = xt + vt and we wish to fit xt, for t = 1, . . . , n, constrained to be smooth, by minimizing

Show that this problem is identical to obtaining xbn t in Problem 6.7(b), with λ = σ2 v/σ2 w, assuming x0 = 0. Hint: Using the notation surrounding equation (6.63), the goal is to find the MLE of Xn given Yn, i.e., maximize log f(Xn|Yn). Because of the Gaussianity, the maximum (or mode) of the distribution is when the states are estimated by xn t , the conditional means.
But log f(Xn|Yn) = log f(Xn, Yn) − log f(Yn), so maximizing log f(Xn, Yn)
with respect to Xn is an equivalent problem. Now, ignore the initial state and write −2 log f(Xn, Yn) based on the model, which should look like (6.210);
use (6.64) as a guide.