= Solution
Introduce a model indicator $M\in\{0,1\}$. In $M=0$, set $\mu=0$, hence $\theta=1$, and retain the two group effects. In $M=1$, include $\mu$ with the prior induced by the gamma baseline. Choose positive prior model <probabilities> $p_0,p_1$ summing to one; the supplied within-model parameter priors alone do not specify these <probabilities>.
Let $r=(a_2,b_2)$, use the same proper prior $p(r)$ in both models, and denote their likelihoods by $L_0(r)$ and $L_1(\mu,r)$. The unnormalized joint model targets are
$$
t_0(r)=p_0p(r)L_0(r),\qquad
t_1(\mu,r)=p_1p(r)p_\mu(\mu)L_1(\mu,r),
\qquad
p_\mu(u)=\frac{b^a}{\Gamma(a)}e^{au-be^u}.
$$
For a birth move, draw an auxiliary $u$ from a positive <probability density function> $q(u)$ and map $(r,u)$ to $(\mu'=u,r'=r)$. The reverse death move simply deletes $\mu$. This is a dimension-matching bijection with absolute Jacobian one. If the birth and death selection <probabilities> are $b_0$ and $d_1$, respectively, the <birth and death moves for Bayesian variable selection> rule gives
$$
\boxed{A_{0\to1}=\min\!\left(1,
\frac{p_1p_\mu(u)L_1(u,r)d_1}{p_0L_0(r)b_0q(u)}\right).}
$$
The reverse acceptance is the reciprocal ratio, capped at one, evaluated at the current $\mu$. Choosing $q=p_\mu$ cancels the added-parameter prior; a data-informed proposal may mix better but must retain its proposal factor. Within each model, also update its continuous parameters to ensure exploration.
For an explicit <likelihood> ratio, with $E$ and $T$ from part (ii),
$$
\frac{L_1(u,r)}{L_0(r)}=\exp\{Tu-(e^u-1)E\}.
$$
This supplies a complete <reversible-jump Markov chain Monte Carlo> move. Its acceptance <probabilities> enforce <detailed balance> between the two dimensions. The lower-dimensional state is a separate point-mass model, not a zero-probability equality test inside the continuous full-model prior.
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