Normalising likelihood for BIC/AIC calculation I am running some model inference using AIC and BIC. My problem is that when I go and calculate the (maximum) loglikelihoods of my models, they are usually really high (range between 4700 and 1400 approx.).
This is mostly because I have a lot of data points, in the order of 20k, and (apparently) decently good models. So the likelihood for each individual point is often > 1 and the sum of their logs is positive and gets quite high.
Now, if I use these values to calculate BIC and AIC and from there the posterior probabilities of my model, I often get a numerical error, because I have very negative BIC/AIC ($<-1000$) and the number I'd get with $\exp(-0.5\text{BIC})$ is just too big for R. This is also the case when I use $\text{BIC}-\text{BIC}_{max}$.
I am considering using normalised likelihood instead of likelihood (i.e. $1/N\cdot \text{likelihood}$). How does this (and sample size in general) affect BIC/AIC and model inference methods? I couldn't really find anything useful to read about this.

I would advise against normalizing the likelihood by the number of observations, since this would make the definitions of the BIC and the AIC irrelevant. AIC/BIC are not arbitrary combinations of a likelihood term and a penalty term: the BIC is an approximation of the model evidence, while the AIC is an unbiased estimator of the Kullback-Leibler divergence between a model and the ground-truth. For instance, using the normalised likelihood would make the approximation $BIC_{\mathcal{M}} \approx -2\log p(\mathcal{D}|\mathcal{M})$ wrong.
Instead of manipulating "wrong" posteriors, you could manipulate "true" BIC scores. Most papers comparing models simply use the values of the BIC without transforming it into a posterior, see for instance Fig. 5 in Pfister, J. P., & Ghosh, A. A priority-based model for smartphone screen touches.
Since you have enough observations to split them between a training and a validation sets, another alternative to compare your models would be to use cross-validation instead of AIC/BIC.
