Flat log-rate marginalization gives the multinomial likelihood (${\int }_0^{\infty }{u}^{n-1}{e}^{-Su}du=\Gamma (n)S^{-n}$)

For positive group totals, the improper prior $du/u$ on an independent Poisson baseline yields the multinomial likelihood kernel after integration. This prior is flat in $\log u$. A proper finite-variance log-rate prior generally introduces an additional coefficient-dependent factor.

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.