Let $f$ be your true distribution, and $g$ the family from which you are trying to fit your data. Then $\theta$, the maximum likelihood estimator of parameters of $g$, is a random variable. You could formulate model selection as finding the distribution family $g$ that minimizes the expected KL divergence between $f$ and $g(\theta)$, which can be written as
Since you are minimizing over $g$, the Entropy($f$) term doesn't matter and you look for $g$ that maximizes $E_x E_y[\log(g(x|\theta(y)))]$.
Let $L(\theta(y)|y)$ be the likelihood of data $y$ according to $g(\theta)$. You could estimate $E_x E_y[\log(g(x|\theta(y)))]$ as $\log(L(\theta(y)|y))$ but that estimator is biased.
Akaike's showed that when $f$ belongs to family $g$ with dimension $k$, the following estimator is asymptotically unbiased
Burnham has more details in this paper, also blog post by Enes Makalic has further explanation and references