Free-knot linear splines: Value of asymptotic theory

Another recent arXival proposes to use a local linear regression as a type of ‘non-parametric’ Bayesian model, although this time the location of the break points is learnt during fitting, which creates a free-knot linear spline model.  The application here is reconstructions of the primordial power spectrum.  Again I don’t generally advocate for such models, but if I did I would want to see a hierarchical prior structure that could promote shrinkage, such as towards a data-driven mean slope and neighbouring slope difference distribution.  Interestingly it seems that the asymptotics literature has something to say on this topic: in particular, that the prior choice on knot locations should not be a Poisson process (such as the independent draws from the Uniform distribution adopted in this arXival), rather it should favour a regular spacing (see Remark 5 in Belitser & Serra).

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