Classical vs Bayesian logistic regression assumptions

Question

In most books, Bayesian logistic regression is usually referred as "better" more advanced than the classical non-Bayesian one. However every time I need to decide which one is most suited to my problem, I end up using the classical one for the simple reason that, if I have understood it correctly, classical logistic regression makes no assumption on the distribution of the independent variables while Bayesian logistic regression does. When looking at the distribution of the predictors for my problems I typically have skewed distribution that don't resemble a normal distribution and that are not easy to model.

Am I missing something about Bayesian logistic regression?

It seems all good when the distributions of the predictors can be easily modeled but this is not always the case.

Answer 1

Yes, I believe you are missing something or it is not clear what "most books" are referring to when they say "Bayesian logistic regression".

Logistic regression is $$ Y_i \stackrel{ind}{\sim} Ber(\pi_i) $$ where $$ \log\left( \frac{\pi_i}{1-\pi_i} \right) = \beta_0+\beta_1 X_{i,1} + \cdots + \beta_P X_{i,P} $$ and $X_{i,1},\ldots,X_{i,p}$ are the independent variables for observation $i$.

Bayesian logistic regression adds the additional assumption $\beta \sim p(\beta)$, e.g. $$ \beta_p \stackrel{ind}{\sim} N(0,v^2)\quad \mbox{for } p=1,\ldots,P.$$ There is no assumption here whatsoever on the distribution of the independent variables.