= Scale identifiability in a gamma random-intercept Poisson model
{title2=$(\beta_0,\tau,\theta)\mapsto(\beta_0-\log c,c\tau,c^2\theta)$}
When $a_{ij}=\exp(\beta_0+\beta^{\mathsf T}x_{ij})$, replacing $U_i$ by $cU_i$ and the fixed intercept by $\beta_0-\log c$ leaves every conditional count mean and the integrated data law unchanged. Thus without fixing the random multiplier's mean, the <gamma random-intercept Poisson model> identifies the marginal intercept $\beta_0+\log\tau$ and relative heterogeneity $\theta/\tau^2$, rather than these three parameters separately. Normalizing $\tau=1$ removes this scale ambiguity.
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