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Assume we have two bivariate random variables $(X_1,X_2)$ and $(Y_1, Y_2)$ and the distribution satisfies $p(y_1,y_2|x_1,x_2)=p(y_1|x_1)p(y_2|x_2)$. I can prove that if $X_1$ and $X_2$ are independent, then $Y_1$ and $Y_2$ are independent. But what about the converse, i.e., if $Y_1$ and $Y_2$ are independent, are $X_1$ and $X_2$ are independent? Thank you for your reply!

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No. Assume $(Y_1,Y_2)$ is independent and independent of $(X_1,X_2)$. Then both sides are equal to $p(y_1)p(y_2)$ but $(X_1,X_2)$ may not be independent.

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