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Is the conclusion(last line) correct? If not, what's wrong with this proof ? (Probability Density Estimation, Bayesian inference)

(Probability Density Estimation, Bayesian inference) $x_1$ and $x_2$ are independent random variables, so $p\left(x_1,x_2\right)=p\left(x_1\right)\cdot p\left(x_2\right)$ and ...
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1answer
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Question about Mutual Information

I am learning about mutual information, and am confused about one of the definitions. Mutual information is defined as $ I(X;Y) = H(X) - H(X | Y) $ where, $$ H(X) = \sum_{x} p(x) \log ...