Why would a smaller AIC than BIC lead to an increased chance of both overfitting and underfitting? I am puzzled by the following statement in my lecture notes

AIC penalty smaller than BIC; increased chance of overfitting 
BIC penalty bigger than AIC; increased chance of underfitting

Is there a typo?

No, there is no typo. The quote makes sense, or at least there is a straightforward interpretation under which it does:

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*The smaller the penalty for model complexity, the more complex the selected model. The more complex the selected model, the more it will overfit. Thus smaller penalty implies greater chance of overfitting.

*The larger the penalty for model complexity, the less complex the selected model. The less complex the selected model, the more it will underfit. Thus larger penalty implies larger chance of underfitting.

There may be another interpretation, though I find it less straightforward in the context of the quote:

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*If we aim for optimal prediction accuracy, we want an efficient selection criterion such as the AIC. Then a consistent selection criterion such as the BIC underfits.

*If we aim for recovering the true model, we want a consistent selection criterion such as the BIC. Then an efficient selection criterion such as the AIC overfits.

However, we aim either for optimal prediction or for recovery of the true model. Then if one criterion overfits, the other one must be optimal (cannot underfit), or if one criterion underfits, the other must be optimal (cannot overfit). Again, I do not find this interpretation as relevant as the first one, but it provides a perspective.
