# Bayesian model selection

August 20, 2017 — July 22, 2019

Frequentist model selection is not the only type, but I know less about Bayesian model selection. What is model selection in a Bayesian context? Surely you don’t ever get some models with zero posterior probability? In my intro Bayesian classes I learned that one simply keeps all the models weighted by posterior likelihood when making predictions. But sometimes we wish to get rid of some models. When does this work, and when not?

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## 1 Sparsity

Interesting special case: Bayesian sparsity.

## 2 Cross-validation and Bayes

There is a relation between cross-validation and Bayes evidence, a.k.a. marginal likelihood - see (Claeskens and Hjort 2008; Fong and Holmes 2020).

## 3 Evidence/marginal likelihood/type II maximum likelihood/Bayes factor

The classic; Worth its own notebook. See model selection by model evidence maximisation.

## 4 Incoming

John Mount on applied variable selection (Mount 2020)

We have also always felt a bit exposed in this, as feature selection

seemsunjustified in standard explanations of regression. Onefeelsthat if a coefficient were meant to be zero, the fitting procedure would have set it to zero. Under this misapprehension, stepping in and removing some variablesfeelsunjustified.Regardless of intuition or feelings, it is a fair question: is variable selection a natural justifiable part of modeling? Or is it something that is already done (therefore redundant)? Or is it something that is not done for important reasons (such as avoiding damaging bias)?

In this note we will show that feature selection

isin fact an obvious justified step when using a sufficiently sophisticated model of regression. This note is long, as it defines so many tiny elementary steps. However this note ends with a big point: variable selectionisjustified. It naturally appears in the right variation of Bayesian Regression. Youshouldselect variables, using your preferred methodology. And youshouldn’tfeel bad about selecting variables.

## 5 References

*Biometrika*.

*Biometrika*.

*Journal of the American Statistical Association*.

*Journal of Statistical Computation and Simulation*.

*Biometrika*.

*The Annals of Statistics*.

*Model Selection*. IMS Lecture Notes - Monograph Series.

*Model Selection and Model Averaging*. Cambridge Series in Statistical and Probabilistic Mathematics.

*The Annals of Applied Statistics*.

*Proceedings of the 32nd International Conference on Machine Learning*.

*Biometrika*.

*Sociological Methodology*.

*Statistica Sinica*.

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*The Annals of Statistics*.

*Journal of the American Statistical Association*.

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*arXiv:1611.01241 [Stat]*.

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*Network: Computation in Neural Systems*.

*Neural Computation*.

*Journal of the American Statistical Association*.

*Nature Genetics*.

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*arXiv:2109.03204 [Math, Stat]*.

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*Journal of the American Statistical Association*.

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*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

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*Journal of the American Statistical Association*.

*Journal of the Korean Statistical Society*.

*Journal of Computational and Graphical Statistics*.

*Frontiers in Applied Mathematics and Statistics*.

*arXiv:1509.09169 [Stat]*.

*Statistics Surveys*.

*Journal of the Royal Statistical Society Series B: Statistical Methodology*.