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Typically, distributions with fat tails, such as the inverse Gamma or the half-Cauchy are used as prior distributions for the variance parameters.

I am trying to understand why do we need a fat-tailed prior for variances. For example, one could simply use a half-normal prior for the variances. What property of the variances make it more suitable to use fat-tailed distributions?

All the resources I could find compare different fat-tailed priors amongst themselves, but I couldn't get anything on why they need to be fat tailed at all.

Now, some of you might say that there is no "correct" and "incorrect" prior and that this should reflect the researcher's beliefs. It is indisputable, however, that there is a tradition of using these fat-tailed priors for variances. I am wondering what created this tradition.

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    $\begingroup$ There absolutely are advantages to using more informative priors, especially for small-data problems. Gelman, for one, is a fan of half-normal priors. Take a look here: statmodeling.stat.columbia.edu/2015/11/07/… $\endgroup$ Commented Sep 25, 2023 at 13:02

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