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An exact Gibbs sampler update is available despite the non-Gaussian intercept: draw ${\theta }^{\prime }$ from the conditional gamma distribution law in part (ii), then setTogether with conditional updates of the two remaining effects, this targets the required joint posterior in the log-intercept coordinates. The log-gamma prior for a Poisson log-intercept includes the change-of-variable factor: with $d\theta /d\mu =e^{\mu }$,Using $(a+T-1)\mu$ instead would omit the Jacobian and give the wrong conditional law for $\mu$.