= Poisson surrogate for a conditional multinomial model
{title2=$\mu_{sl}=e^{\gamma_s+x_s^T\beta_l}$}
Independent <Poisson distributions> with a separate nuisance intercept in every stratum become <multinomial distributions> after conditioning on their totals. Profiling the intercept gives $e^{\widehat\gamma_s}=N_s/\sum_l e^{x_s^T\beta_l}$. Substitution produces the same parameter-dependent <log-likelihood> as the conditional multinomial model. Both approaches fit $\widehat\mu_{sl}=N_s\widehat p_{sl}$. This exact equivalence differs from approximating a single binomial event count by a rare-event <Poisson distribution>.
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