Gaussian log-rate correction to the Poisson trick ($R(S)={\mathbb{E}}_{\Gamma (n,1)}{e}^{-(\log V-\log S{)}^2/(2s^2)}$)

A mean-zero normal distribution prior of variance $s^2$ on the log baseline multiplies the multinomial kernel by the displayed correction, up to a constant independent of coefficients. The factor is not constant in $S$, so the posterior equivalence is approximate. Dominated convergence makes it tend to one as $s^2$ tends to infinity.