I'm wondering what parameter BIC estimates. It seems that AIC is estimating the cross entropy of the estimated model and the true model, and asymptotically is estimating the out of sample entropy loss. If you divide AIC by 2n it seems that this scaled AIC ($\hat{L}/n - p/n$, where $\hat{L}$ is the maximum likelihood of the observed data, p is the number of parameters, and n is the sample size) is estimating the following:
$\int g(x) log\left( f(x|\theta)\right)$
Akaike mentions this fact in his 1974 paper.
But BIC has a different penalty term ($\log(n)p$ instead of $2p$) so I am wondering what parameter (if any) BIC is estimating.
Additionally, in the limit, it appears that the penalty terms for AIC and BIC are both approaching zero, so in the limit I would expect them to both estimate the same parameter. But AIC and BIC have different asymptotic properties (consistency of BIC and efficiency of AIC). Any insight on this would be helpful.
References:
Akaike, Hirotugu. "A new look at the statistical model identification." IEEE transactions on automatic control 19.6 (1974): 716-723.