Past exam of the simulation and statistical programming course of the University of Oxford / 2021 / 1 / c

An importance weight has expected value ${\mathbb{E}}_q[w(Z)]=\int p(z)dz=1$. From the ratio in (a),The supplied normal distribution transform applies with quadratic coefficient $-(1-s^{-1})$, giving each factor $(2s-1{)}^{-1/2}$. ThusFor $s>1$, the base $A:=s/\sqrt{2s-1}$ exceeds one because $s^2-(2s-1)=(s-1{)}^2>0$: importance weight variance increases exponentially with dimension. For $s=1$, the importance weight is identically one and its variance is zero. This calculation concerns the importance weight, rather than the weighted integrand.