We derive a new efficient algorithm for optimal spline smoothing as the conditional expectation of a stochastic process observed with noise, using the stochastic model of Wahba (J. Royal Statist. Soc. Ser. B, 40 (1978), pp. 364–372). The conditional expectation is computed by expressing the process in state space form and using the filtering and smoothing results in Ansley and Kohn (Annals Statist., 11 (1985), pp.1286–1316). We show how to use our algorithms to estimate the smoothness parameter and how to obtain Bayesian confidence intervals for the unknown function and its derivatives. Algorithms based on other stochastic models are compared to ours, and a stochastic derivation is given for Reinsch’s (Numer. Math. 10 (1967), pp. 177–183) algorithm for polynomial splines.