Poisson trick

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 ${\hat{u}}_g=n_g/S_g(\beta )$.

Substituting this into the likelihood leavesThis 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 $u_g\downarrow 0$; 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.

For independent Poisson distribution cells with means $u{e}^{{\eta }_j(\beta )}$, write $n={\sum }_j{y}_j$ and $S(\beta )={\sum }_j{e}^{{\eta }_j(\beta )}>0$. Their likelihood, apart from count-factorial constants, isAn improper prior flat in $\log u$ has measure $du/u$ on $u>0$. When $n>0$, integration givesso the remaining kernel in $\beta$ is ${\prod }_j[e^{{\eta }_j(\beta )}/S(\beta ){]}^{y_j}$, proportional to the multinomial likelihood. The gamma function integral is the exact reason for the equivalence.

This is proportionality of parameter kernels after an improper-prior calculation; it does not supply a normalized model evidence. For a zero group total, the baseline integral diverges at $u=0$, so the same improper-prior marginalization is invalid even though the conditional multinomial likelihood is constant. A proper prior on the baseline generally changes the kernel. The Gaussian log-rate correction to the Poisson trick makes this additional factor explicit.

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.