Why do we use a criterion like AIC for Copula model selection?

Question

If we look at the AIC formula:

AIC = -2*log(ML) + 2k

where k is the number of parameters in the model and is considered as the 'penalizing term' for complexity or over-fitting. Does this assumption of complexity/over-fitting actually apply to copulas? In terms of other models, like a simple linear regression model, I understand that adding more parameters always increases your model's performance so it makes sense to penalize it. But if I fit two copula models to my data, one that has a single parameter (like Clayton copula) and another that has two parameters (like BB1 copula), does that mean the BB1 model should be penalized? What is the intuition behind it specifically in copula models?

Answer 1

The AIC represents and estimate of the Kullback Liebler divergence between the model that generated the data (true model) and the proposed parametric model. Thus, the model with the smallest AIC means that such model is the closest, in the KL divergence sense, to the true model.

https://en.wikipedia.org/wiki/Akaike_information_criterion#How_to_apply_AIC_in_practice

However, you need to be careful about the selection of non nested models in terms of AIC as some models may reflect some features of the data while others reflect different properties of the data. For example, let's say that the true model is heavy tailed and asymmetric and that you fit a light-tailed asymmetric model, and a symmetric heavy tailed model. It may be the case that one of them is favoured by the data in terms of AIC, but it does not mean that it captures all the features of the data. So, some discretion and reflection is recommended when using AIC, as it is certainly not an automatic criterion.