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The full conditional log probability density functions of the effects are, up to constants,These are Gaussian-prior Poisson log-effect conditional probability density functions. The exponential likelihood term prevents a standard normal distribution conjugate update. A Metropolis-within-Gibbs algorithm conveniently updates one effect at a time without computing its conditional normalizing constant.

For example, propose $a_2^{\prime }=a_2+s_{\alpha }Z$ with $Z\sim N(0,1)$, and accept withUse an analogous normal distribution random-walk proposal for $b_2$. The proposal is symmetric and lives on the appropriate unrestricted real parameter space, so no proposal ratio is needed. It makes local moves compatible with a unimodal conditional probability density function. Tune the positive scale in a pilot or freeze it after warm-up; a scale far too large causes rejection and a scale far too small moves slowly.

A useful scale can be inferred from the conditional curvature: ${\ell }_{\alpha }^{\prime \prime }(a_2)=-I\theta (1+e^{b_2})e^{a_2}-{\sigma }_1^{-2}<0$, with the analogous formula for $b_2$. A normal proposal near the conditional mode with variance approximately $-1/{\ell }^{\prime \prime }$ is another sensible choice, but an asymmetric or state-dependent proposal must include its full reverse-to-forward proposal-density ratio. Metropolis–Hastings is convenient, not logically mandatory; these concave conditionals also admit other specialized samplers.