jessicata comments on A possible training procedure for human-imitators

[jessicata]

What I mean is: compute ${\hat{E}}_q\left[\right[f\left(A\right)=x\left]p\right(A\left)/q\right(A\left)\right]$, which is a probabilistic lower bound on $P_p\left(f\right(A)=x)$.

The variational score gives you a somewhat worse lower bound if $q$ is different from $p\left(A|f\right(A)=x)$. Due to Jensen’s inequality, $$E_q\left[\log \right(\left[f\right(A)=x]p\left(A\right)/q\left(A\right)\left)\right]\le \log E_q\left[\right[f\left(A\right)=x\left]p\right(A\left)/q\right(A\left)\right]\le \log P_p\left(f\right(A)=x)$$

It probably doesn’t make a huge difference either way.