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I'm stuck trying to find the maximum entropy probability distribution taking into account a joint distribution. Basically, I want to find the maximum entropy expression for $p(x,y)$ when the marginal distributions $p(x)$ and $p(y)$ are known.

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Well $H(X,Y)\leq H(X)+H(Y)$ with equality if and only if $X$ and $Y$ are independent. So pick $p(x,y)=p(x)p(y).$

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