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My understanding: choosing knots for a B-spline can be an arcane task filled with guessing and eye-balling. Penalized B-splines attempt to do away with the choice of knot picking, fitting a spline by:

1) Using many basis B-splines (yes, that sounds oxymoronic since bases over a vector space have the same number of elements -- but the idea is that we consider a linear combination of many B-splines)

2) Penalizing the coefficients of these basis B-splines using some penalty while fitting them to some data set.

Assuming that my understanding above is correct: is there any other point to using penalized B-splines?

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This is basically correct, but I think you're under-stating it a bit. Point (2) is a really big deal, though, since you can abstract away worrying about how and where to place your knots: place a lot of knots, then choose the correct level of penalization. This can be done with cross-validation or alternative methods which directly optimize a fitness criterion.

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    $\begingroup$ you don't even have to cross-validate, modern packages for estimation of penalized splines can also directly optimize ML or REML criterion (e.g. mgcv in R) $\endgroup$
    – adibender
    May 21, 2019 at 16:14
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    $\begingroup$ @adibender Good point. Revised. $\endgroup$
    – Sycorax ♦
    May 21, 2019 at 16:24
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I think that another important point is related to the interpolation and extrapolation properties of Penalized B-splines (aka P-splines).

A really nice discussion can be found here: Splines, knots, and penalties by Paul H. C. Eilers, Brian D. Marx https://doi.org/10.1002/wics.125

Citing the authors,

When interpolating, the B-spline coefficients form a sequence of degree $2d − 1$, when extrapolating, of degree $d−1$. Thus, when $d = 2$, we get cubic interpolation and linear extrapolation

where $d$ is the order of the difference penalty.

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  • $\begingroup$ would you please put the full reference to the paper that you linked above in a comment? The link not working anymore... $\endgroup$
    – Arrigo Benedetti
    Nov 2, 2022 at 1:01
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    $\begingroup$ @ArrigoBenedetti I edited the post. The reference should be OK now $\endgroup$
    – Gi_F.
    Nov 2, 2022 at 7:23

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