= Constrained Poisson and multinomial likelihood equivalence
For cell means $\mu_j$ with $\sum_j\mu_j=n$, the independent <Poisson distribution> likelihood and the <multinomial distribution> likelihood with probabilities $\mu_j/n$ differ by a parameter-independent factor. Thus a <maximum-likelihood estimator> for the unrestricted Poisson model whose fitted total is $n$ is also an estimator for the constrained multinomial model. Equality of named parameter estimates presumes identifiability and uniqueness; without these, the conclusion concerns the sets of maximizers. A free intercept in a log-linear Poisson model enforces the fitted-total equation.
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