= Poisson trick
{c}
The <Poisson trick> obtains a fixed-total <multinomial likelihood> by fitting <independent> <Poisson distribution> cells with a free baseline rate for each group. Write $\mu_{gj}=u_g e^{\eta_{gj}(\beta)}$ with $u_g>0$ and $S_g(\beta)=\sum_j e^{\eta_{gj}(\beta)}$. For an observed positive group total $n_g$, maximizing its Poisson likelihood over $u_g$ gives $\widehat u_g=n_g/S_g(\beta)$.
Substituting this into the likelihood leaves
$$
L_g^{\mathrm{prof}}(\beta)\propto
\prod_j\left[\frac{e^{\eta_{gj}(\beta)}}{S_g(\beta)}\right]^{y_{gj}}.
$$
This is the conditional <multinomial likelihood> in $\beta$. The omitted factor depends on the observed counts, not on $\beta$, so profile <maximum-likelihood estimators> and <likelihood-ratio tests> agree when they are otherwise well-defined. The group baseline must be free; constraints or penalties on it can break the equivalence.
For a group with counts $3,1$ and saturated cell predictors, both approaches fit proportions $3/4,1/4$. If a group total is zero, profiling over strictly positive $u_g$ has its supremum at the excluded boundary $u_g\downarrow0$; the conditional likelihood from that group is constant and supplies no information about its proportions. <Flat log-rate marginalization gives the multinomial likelihood> describes a different operation, integration over the baseline rather than maximization.
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