 Solution

Both criteria are used for model selection. AIC estimates relative out-of-sample predictive or Kullback-Leibler risk and is attractive when prediction is the main aim. BIC approximates a log Bayes factor under regular fixed-dimensional models and is consistent for selecting a true finite-dimensional model when one is present. Their penalties differ by $2k$ versus $k\log n$. For $n>e^2$, BIC penalizes each additional parameter more strongly and therefore tends to select smaller models.