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I consider a target distribution $\mu$, a proposal distribution $\nu \gg \mu$ (both probability distributions), $w= d\mu/d\nu$, $f$ a bounded function.

The question is : prove that $$ Var_{\nu} (fw) = \mu(f)^2 \nu\left[\left( \frac{|f|w}{\mu(|f|)}-1 \right)^2\right] $$ I keep messing with the definitions.

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