Last year I wrote a series of posts comparing frequentist and Bayesian approaches to various problems:
Here [perma] I am going to dive into an important topic that I've not yet covered: model selection. We will take a look at this from both a frequentist and Bayesian standpoint, and along the way gain some more insight into the fundamental philosophical divide between frequentist and Bayesian methods, and the practical consequences of this divide.
My quick, TL;DR summary is this: for model selection, frequentist methods tend to be conceptually difficult but computationally straightforward, while Bayesian methods tend to be conceptually straightforward but computationally difficult.
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