Natural Cubic Splines: Optimization and Penalization

This lecture covers the optimization and penalization techniques related to natural cubic splines, including the use of basis functions, roughness penalties, and equivalent formulations. The instructor explains the process of constructing natural cubic splines, integrating second derivative functions, and the optimality of these splines. The lecture also delves into the concept of roughness penalties, the balance between fidelity to data and smoothness, and the implications of using prior distributions in Bayesian inference and mixed effects models.