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I am reading this paper which rewrites ELBO. I am stuck in verifying the mathematics used for doing the rewriting. Essentially, the paper writes the KL term involved in ELBO as follows (equations 13 through 15 using the equivalencies defined in equation 9-10). The confusion starts right from the first line where the authors write: $$\frac{1}{N}\sum_{n=1}^{N} \text{KL}(q(z_n|x_n)||p(z_n)) = \sum_{n=1}^{N} q(z_n,x_n)\log\frac{q(z_n,x_n)}{p(z_n,x_n)}$$ I can see how using $$q(z_n|x_n)=q(z_n,x_n)q(x_n)=q(z_n,x_n)\times\frac{1}{N},\\p(z_n|x_n)=p(z_n,x_n)p(x_n)=p(z_n,x_n)\times\frac{1}{N}$$ we can rewrite the LHS to, $$\frac{1}{N}\sum_{n=1}^{N} \text{KL}(q(z_n|x_n)||p(z_n)) = \sum_{n=1}^{N}\int_z q(z_n,x_n)\log\frac{q(z_n,x_n)}{p(z_n,x_n)}$$ but don't understand why they don't have $\int_z$ in their math when expanding the KL term.

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