Hi Nikos!
AIC (and BIC) are not Bayesian quantities. Taken from wikipedia:
Suppose that we have a statistical model of some data. Let k be the number of estimated parameters in the model. Let \hat{L} be the maximum value of the likelihood function for the model. Then the AIC value of the model is the following.[1][2]
\text{AIC} = 2k-2\log(\hat{L})
…as you see, AIC relies on the maximum likelihood estimator. From a Bayesian view the maximum of the likelihood function is not that interesting, as we work with the whole posterior distribution. For model assessment you can use WAIC, or better yet LOO – which you already use! :)
What do you want to use AIC or BIC for? Comparisons with MLE methods?