  The P-splines of Eilers and Marx (1996) combine a B-spline basis with a
discrete quadratic penalty on the basis coefficients, to produce a reduced rank
spline like smoother. P-splines have three properties that make them very
popular as reduced rank smoothers: i) the basis and the penalty are sparse,
enabling efficient computation, especially for Bayesian stochastic simulation;
ii) it is possible to flexibly `mix-and-match' the order of B-spline basis and
penalty, rather than the order of penalty controlling the order of the basis as
in spline smoothing; iii) it is very easy to set up the B-spline basis
functions and penalties. The discrete penalties are somewhat less interpretable
in terms of function shape than the traditional derivative based spline
penalties, but tend towards penalties proportional to traditional spline
penalties in the limit of large basis size. However part of the point of
P-splines is not to use a large basis size. In addition the spline basis
functions arise from solving functional optimization problems involving
derivative based penalties, so moving to discrete penalties for smoothing may
not always be desirable. The purpose of this note is to point out that the
three properties of basis-penalty sparsity, mix-and-match penalization and ease
of setup are readily obtainable with B-splines subject to derivative based
penalization. The penalty setup typically requires a few lines of code, rather
than the two lines typically required for P-splines, but this one off
disadvantage seems to be the only one associated with using derivative based
penalties. As an example application, it is shown how basis-penalty sparsity
enables efficient computation with tensor product smoothers of scattered data.
