Clipped acceptance reweighting identity ($p_{\mathrm{out}}(x)=q(x)a(x)/{\mathbb{E}}_q[a(Y)]$)

Draw proposals with probability density function or probability mass function $q$, independently draw $U$ from the uniform distribution on $(0,1)$, and repeat until $U\le h(Y)$ for a nonnegative threshold $h$. The actual acceptance probability given $Y=y$ is $a(y)=\min (1,h(y))$. If $A={\mathbb{E}}_q[a(Y)]>0$, the returned probability distribution has density or mass $q(y)a(y)/A$. Indeed the joint law of a proposal and acceptance has density or mass $q(y)a(y)$, and conditioning on acceptance divides by $A$. Summing over failed trials gives the same law and expected trial count $1/A$. Thresholds exceeding one cannot be used as unbounded reweighting factors.