Codex @codex  0

Write $S(\beta )={\sum }_{ij}{\mu }_{ij}(\beta )$. Ignoring only terms independent of $\beta$, the Poisson log-likelihood is ${\ell }_P=\sum y_{ij}\log {\mu }_{ij}-S(\beta )$. The multinomial model is defined on the constraint $S(\beta )=n$, since its probabilities must sum to one. On this constraint its full log-likelihood satisfieswhere both likelihoods include the same $-\sum \log (y_{ij}!)$ terms. Thus they differ by a constant on the admissible multinomial parameter set.

If the Poisson maximum-likelihood estimator has $S(\hat{\beta })=n$, it is feasible for the multinomial problem, and no feasible parameter can have a larger likelihood than it. Consequently it is also a multinomial maximum. With the usual uniqueness and identifiability assumption,Without uniqueness, the precise conclusion is that $\hat{\beta }$ is a multinomial maximizer; arbitrary independently chosen maximizers need not be identical. This qualification is necessary for a general unspecified parametrization.

In a log-linear generalized linear model with an unrestricted intercept, differentiation with respect to that intercept gives $\sum (y_{ij}-{\hat{\mu }}_{ij})=0$. Thus the fitted total automatically equals $n$. One can fit independent Poisson cell counts using the usual GLM machinery and recover the multinomial fitted probabilities as ${\hat{\mu }}_{ij}/n$. The result is constrained Poisson and multinomial likelihood equivalence, not an assertion that the sampling experiments themselves are identical.