THE FRAMEWORK:
Let $X_1$ be an observation from a normal random variable with mean zero and variance $\sigma^2$ and lets call the PDF $f(x)$.
I want to minimize the Kullback Liebler Information criterion between a PDF $g(x, \beta) $ of a zero mean normal random variable with variance $\beta \sigma^2$ and $f(x)$. The minimization is with respect to $\beta$.
The Kullback Liebler Information criterion is defined as t
$$I(f: g, \beta):= E [ \log(f(X_1)/g(X_1, \beta) )]$$
THE MOTIVATION:
The motivation for doing this is that Akaike showed that the maximum likelihood estimator $\hat{\beta}$ of a model that assumes $g$ as the parametric distribution generating the observations is a natural estimator for the value $\beta^*$ that minimizes the Kullback Liebler information criterion.
THE PROBLEM:
I wanted to do this simple calculation because I expected $\beta^* = 1$ (in this way the two distributions would be equal so their "distance" is minimized). But performing the computations I obtain
$$ E \left[ \frac{1}{2} \log \beta X_1^2 - \frac{1}{2} \log \sigma^2 - \frac{\beta X_2^2 + \sigma^2}{2 \sigma^2 \beta} \right] $$
and it seems to me that the $\beta^* $ that minimizes this tends to minus infinity. Where is my mistake?