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Suppose $p(x)$ approximates the density of interest $q(x)$. Then $$\int f(x) q(x) = \int f(x) \left(\frac{q(x)}{p(x)} \right) p(x) \ dx = E_{p(x)} f(x) \left(\frac{q(x)}{p(x)} \right)$$

Why don't the $p(x)$'s cancel in the second equality?

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The notation $E_{p(x)}$ is illogical, instead the RHS should read $E_p(fq/p)$. –  Did May 5 '12 at 6:01
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1 Answer 1

They do cancel in the middle phrase. In fact, that's how you know that the first statement is equal to the second statement. To get from the first to the second, they multiplied by $p(x)/p(x) = 1$, which doesn't change anything. They do this so that they can write it in the form of the last statement.

Without knowing the context, it's difficult to say more.

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