Ye Official Prior Choice Recommendations dictate that when specifying weakly informative priors for logistic regression coefficients, “[n]ormal distribution is not recommended as a weakly informative prior, because it is not robust”. Student-t distributions with df from 3 to 7 are recommended instead.
Then it goes on to say that the Gaussian distribution is fine if the prior is intended to be informative rather than just weakly so. This is fairly mystifying to me as well.
I do understand how the Gaussian distribution is “less robust” than a student-t one when y is continuous. For example, if you’re using a Gaussian distribution to model male heights in centimeters (with true values e.g. of \mu = 180, \sigma = 6) and your sample includes an outlier suffering severe gigantism with a height of 300cm, then a Gaussian model will shift the estimated location and scale much farther up in response to the outlier than a student-t model, which will merely thicken the tail while leaving the estimated location and scale nearly unaffected.
But I don’t understand how this applies to prior specifications for logistic regression coefficients. When I fit binomial models to mock data with a wide range of different logits corresponding to the \beta's, a Gaussian prior with a given Scale always imposes more shrinkage on large logits than a student-t prior with the same Scale. Thus, I don’t see in what sense the student-t prior can be regarded as “more robust” for these parameters. The mind boggles.