= Solution
The full conditional log <probability density functions> of the effects are, up to constants,
$$
\begin{aligned}
\ell_\alpha(a_2)&=Ra_2-I\theta(1+e^{b_2})e^{a_2}-\frac{a_2^2}{2\sigma_1^2},\\
\ell_\beta(b_2)&=Cb_2-I\theta(1+e^{a_2})e^{b_2}-\frac{b_2^2}{2\sigma_2^2}.
\end{aligned}
$$
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 $a_2'=a_2+s_\alpha Z$ with $Z\sim N(0,1)$, and accept with
$$
\boxed{\min\{1,\exp(\ell_\alpha(a_2')-\ell_\alpha(a_2))\}.}
$$
Use 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: $\ell_\alpha''(a_2)=-I\theta(1+e^{b_2})e^{a_2}-\sigma_1^{-2}<0$, with the analogous formula for $b_2$. A normal proposal near the conditional mode with <variance> approximately $-1/\ell''$ 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.
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