More on Splines: Penalised Likelihood and Natural Cubic Splines

This lecture delves into penalised likelihood in the context of natural cubic splines, aiming to balance fidelity to data and smoothness of the estimated function. The unique explicit solution to this problem is explored, showcasing the construction of natural cubic splines with specific properties. The lecture also covers the derivation of a 'magic' function related to integrated Brownian motion, the basis for natural cubic splines, the optimality of spline interpolation, and the relationship between smoothing and interpolation. Through step-by-step explanations, the lecture demonstrates the process of finding the optimal spline function that minimises a curvature functional, emphasizing the importance of smooth interpolation in achieving this goal.