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I recently read in a paper comparing the Full Bayesian approach to the Empirical Bayes the following : "the FB method can accom- modate distributions such as the hierarchical Poisson-Gamma distribution and the Poisson-LogNormal distribution" (Persaud et al., 2010) Does this mean that performing Full Bayes with informative priors using logistic regression violates the Full Bayes assumptions? From what I have already seen in the forum and other books there are no extra assumptions compared to the frequentist logistic regression, is there something that I understood wrong? Sorry if my question is not very clear I am really new in Bayesian statistics. P.S. Any recommendations on books/papers regarding the application of Bayesian Logistic Regression with informative priors (potentialy using a "historical" dataset) are more than welcome!

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Empirical Bayes is different to the "Fully Bayesian" solution, where (i.e. in empirical Bayes) you use the data to decide what parameters you want to use for your prior distribution.

There are additional assumptions in Bayesian logistic regression compared to the frequentist setting, namely that you decide on a prior for you model parameters. The effect of adding a prior on your regression coefficients is similar to that of penalised likelihood methods in a frequentist setting (ridge regression, etc.).

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  • $\begingroup$ Thank you for your answer! Are there any restrictions regarding the prior distributions that can be used in a bayesian logistic regression? $\endgroup$ – anna.sour Nov 6 '18 at 17:47
  • $\begingroup$ It has to lead to a distribution over the parameters that actually is a probability distribution (the prior itself does not need to be one). Other that, some prior distributions are harder to work with than others, but in principle you can use any choice that leads to a probability distribution over your parameters. $\endgroup$ – Simon Boge Brant Nov 8 '18 at 11:12
  • $\begingroup$ Many thanks Simon. Do you have any papers/books to suggest on this topic ? $\endgroup$ – anna.sour Nov 8 '18 at 16:08

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