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A Gamma–Poisson hierarchical model uses $Y_i\mid {\theta }_i\sim \mathrm{Poisson}(t_i{\theta }_i)$, ${\theta }_i\mid b\sim \mathrm{Gamma}(a,b)$, and $b\sim \mathrm{Gamma}(c,r)$, with gamma distributions in shape-rate convention. Conditional independence gives full conditional distributions ${\theta }_i\mid b,y\sim \mathrm{Gamma}(a+y_i,b+t_i)$ and $b\mid \theta ,y\sim \mathrm{Gamma}(c+na,r+{\sum }_i{\theta }_i)$.

For conditionally independent paired counts with means ${\lambda }_i$ and ${\lambda }_i{e}^{\beta }$, condition on every pair total $N_i=Y_{i0}+Y_{i1}$. The joint conditional likelihood removes all pair-specific rates, leaving the displayed independent binomial distributions. A zero-total pair contributes no information. With unrestricted fixed baseline rates, this conditional likelihood has the same parameter-dependent part as the ordinary profile likelihood; a specified Gamma–Poisson hierarchical model can retain additional information by imposing structure on those rates.

The joint density factors as $p(\psi ){\prod }_{j}p({\theta }_j\mid \psi )p(y_j\mid {\theta }_j)$. In the full conditional distribution of ${\theta }_i$, all factors not involving ${\theta }_i$ cancel. The remaining likelihood function and exponential distribution prior giveIn shape–rate convention this is $\boxed{{\theta }_i\mid \psi ,y\sim \mathrm{Gamma}(y_i+1,1+\psi )}$, a special case of the Gamma–Poisson hierarchical model. It depends only on the node's observed child $y_i$ and parent $\psi$, as required for its Gibbs sampler update.

For pair $i$, use conditionally independent counts $Y_{i0}\sim \mathrm{Pois}({\lambda }_i)$ and $Y_{i1}\sim \mathrm{Pois}({\lambda }_i{e}^{\beta })$, with a common effect $\beta$ and a separate positive baseline rate for each physician. Equivalently the linear predictor is ${\alpha }_i+\beta x$, with ${\alpha }_i=\log {\lambda }_i$. This Poisson regression controls each physician's stable baseline propensity.

Condition on all twelve pair totals $N_i=Y_{i0}+Y_{i1}$. The paired Poisson conditional likelihood has independent factorsThis eliminates every ${\lambda }_i$ exactly. It is more useful as a nuisance-elimination device than the unpaired calculation, which removes only one common baseline: many pair-specific rates are otherwise estimated from just two measurements each.

A possible concern is loss of information through conditioning on totals that are not ancillary for $\beta$, especially with sparse pairs; a zero-total pair contributes no conditional information. There is an important qualification in this exact fixed-baseline Poisson model. Maximizing over ${\lambda }_i$ gives ${\hat{\lambda }}_i=N_i/(1+e^{\beta })$, and substitution leaves precisely the same $\beta$-dependent factor as the conditional likelihood. Hence there is no extra loss relative to ordinary profiling in this particular model; one should not automatically assert an incidental-parameter bias in its estimate of $\beta$.

Additional baseline observations, a calibrated distribution of baseline rates, or informative hierarchical constraints can retain information in the pair totals. A Gamma–Poisson hierarchical model is one possible structured model, but merely giving it a completely free common mean does not automatically create extra information about $\beta$: that mean can absorb the factor $1+e^{\beta }$ in the totals. Such remedies trade exact nuisance elimination for additional data or assumptions. Finally, paired sampling alone does not establish conditional Poisson independence or equidispersion: remaining within-physician dependence or overdispersion should be checked, and an appropriate joint count model or physician-level robust variance used if needed. The common multiplicative-effect assumption is also part of the analysis.

Use the shape-rate convention for every gamma distribution, with $a,c>0$. Let $A_i=a+y_i$ and $s={\sum }_i{\theta }_i$. The Poisson distribution likelihood and the Gamma–Poisson hierarchical model giveAll factors involving the coordinate being updated must be retained, including $b^{na}$ from the conditional gamma distributions. The full conditional distributions areFor a blocked Gibbs sampler, draw $b$ from the second full conditional distribution, then redraw all the ${\theta }_i$ independently from the first ones. Repeating these two steps preserves the joint posterior distribution; observing $\theta$ after each complete sweep gives the Markov chain used in the next part. Standard gamma distribution sampling works for noninteger as well as integer shapes. The positive observation-period convention $t_i>0$ is used for the subsequent geometric drift condition.