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Is "Information" somehow Related to "Variance"?

Why is the "most informativeness property" useful when selecting probability distributions? It isn't. It relates to the principle of maximum entropy, that ... is based on the premise that ...
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Prove that the maximum entropy distribution with a fixed covariance matrix is a Gaussian

We can also verify the above result by solving the constrained optimization problem. For simplicity, let's prove for one dimensional random variable $X$ first (with $\Sigma=[\sigma^2]$ fixed, with $V(...
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