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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 $E_p(fq/p)$.2012-05-05

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