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The Poisson trick introduces ${\lambda }_i=\log {\mu }_{i1}$. With an exactly flat log-rate prior, the baseline prior measure is $d{u}_i/u_i$. For $n_i>0$,Thus flat log-rate marginalization gives the multinomial likelihood:Alternatively, independent Poisson counts conditional on their total are exactly multinomial distributions with probabilities $e^{{\eta }_{ik}}/S_i$. Locally uniform coefficient priors make the coefficient posterior proportional to this likelihood within their flat region.

The printed BUGS prior is a proper normal distribution on ${\lambda }_i$, with variance $s^2=10^5$, since BUGS uses precision as its second argument. It is broad but not exactly flat. Substitution $v=S_i{u}_i$ shows that, apart from a coefficient-independent constant, the integrated likelihood is the multinomial kernel times the Gaussian log-rate correction to the Poisson trickIt depends on the coefficients through $S_i$. Dominated convergence gives $R_i\to 1$ as $s^2\to \infty$, so the finite-variance code yields an approximation to the multinomial posterior, not exact equality. Large log rates can make prior sensitivity relevant.

The cell-wise factors form a standard Poisson regression, convenient for BUGS. They avoid an explicitly constrained count vector and permit scalar log-concave updates for intercepts and coefficients; an exactly flat-log-rate implementation also gives gamma conditional baseline rates. This can be computationally efficient, although the actual log-normal baseline prior is not gamma-conjugate and introduces $I$ nuisance intercepts. Efficiency depends on the update scheme.

This is the appropriate command for the full contingency table. It fits a log-linear model to the nine cell counts, with logarithmic link function and no interaction:With baseline constraints on $g_i,d_j$, there are five parameters. The factorized means express independence of treatment group and damage category. The maximum-likelihood estimates areAlthough the experiment fixes each row total at 30, conditioning these independent Poisson random variables on those totals produces the appropriate row-wise multinomial distribution. The Poisson trick therefore gives the correct likelihood inference for this independence log-linear model for a two-way contingency table; the margins are nuisance parameters. It tests whether treatment changes the distribution of damage, rather than modelling a binary outcome.

Write $y_{i+}={\sum }_j{y}_{ij}$ and ${\mu }_{i+}={\sum }_j{\mu }_{ij}$. Under the model in part (a),and $p_j$ is independent of the price category $i$. The Poisson-multinomial conditioning identity givesMore explicitly, the likelihood function for row $i$ factors asThe first factor is the likelihood of the row total and the second is the conditional multinomial likelihood. The free price effect ${\alpha }_i$ makes the first factor maximal at ${\hat{\mu }}_{i+}=y_{i+}$, while maximizing the remaining factors givesConsequently both fits have the same maximum-likelihood fitted valueHere every row total is $30$, the column totals are $31,28,31$, and $y_{++}=90$, so every fitted row isThis factorization is the Poisson trick.

Letand condition on $N_g=n_g$. The corresponding model for each treatment group isFor fit1, the additive Poisson means implyso the outcome probabilities are common to all three treatments. For fit2,so each treatment may have its own outcome probabilities.

The Poisson trick is Poisson-multinomial conditioning: independent Poisson cell counts, conditional on their total, are multinomial with probabilities proportional to their means. With free treatment main effects, profiling the Poisson likelihood over the group totals gives the multinomial likelihood up to a parameter-independent factor. The Poisson regressions therefore fit these multinomial models and give the same likelihood-ratio comparisons for the outcome parameters.