Let's say I have measurements $Y$ which are all positive, and the distribution seems to be somewhat skewed. I'm modelling $Y$ in GLM framework. Now I could set my GLM using different distributional assumptions for $Y$, i.e. I could set family as normal or gamma, or I could log-transform $Y$ and use normal distribution for that. The problem is, how do I compare these models and figure out which is the best? My thinking is that I cannot use AIC or BIC because of different distributions (or in log-transformation case different response variable values). S what wold be the correct way? Only thing I have come up is checking the histograms of residuals and seeing what they look.
Edit: Thinking more, if likelihood is the probability of the data given the model, and if all models have same number of parameters, I can directly compare the log-likehoods, which must contain all the constants also. Can somebody confirm this?