When you have a discrete number of predictors or terms in your model, you need to choose how many to use, based on the amount of data you have, and how well it is explained by these various models. This is a kind of dual to statistical learning theory where you hope to quantify how complicated a model you should bother fitting to a given amount of data.

If your predictors are discrete and small in number, you can do this in the traditional fashion,
by *stepwise model selection*, and
you might discuss the degrees of freedom
of the model and the data.
If you are in the luxurious position of having a small tractable number of parameters and the ability to perform controlled trials, then you do
ANOVA.

When you have regularisation parameters, we tend to phrase this as
smoothing
and talk about *smoothing parameter selection*, which we can do in various ways.
I’m fond of degrees-of-freedom penalties because they aren’t worse than cross-validation, but much quicker.
However, I’m not yet sure how to make that work in
sparse regression.

Multiple testing is model selection writ large, where you can considering many hypothesis tests, possible effectively infinitely many hypothesis tests, or you have a combinatorial explosion of possible predictors to include.

TODO: document connection with graphical models and thus conditional independence tests.

## Consistency

If the model order *itself* is the parameter of interest, how do you do consistent inference of that?

An exhausting, exhaustive review of various model selection procedures with an eye to consistency, is given in RaWu01.

## Cross validation

See cross validation.

## Under sparsity

Fiddly. See sparse model selection.

## Hyperparameter search

How do you choose your hyperparameters? (nb hyperparameterers might not always be about model selection per se, but rather some kind of convergence rate. Anyway.)

Turns out you can cast this as a bandit problem, or a sequential optimisation problem.

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