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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 areFor a birth move, draw an auxiliary $u$ from a positive probability density function $q(u)$ and map $(r,u)$ to $({\mu }^{\prime }=u,r^{\prime }=r)$. 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 givesThe 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),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.

The cross-model target must include a model-order prior ${\rho }_k$. Write its joint, unnormalized density asHere the prior densities are normalized within each model. In particular their $k$-dependent determinants and powers of $2\pi$ cannot be dropped merely because the fixed-model posterior in part (i) was given up to proportionality.

For explicit birth and death moves for Bayesian variable selection, choose a birth with probability $b_k$, generate $u$ from a density $g_k(u\mid a,v)$, append $u$ as the new covariate coefficient, and leave $v$ unchanged. The map from $(a,v,u)$ to $(a^{\prime },v^{\prime })=((a,u),v)$ is invertible under the reverse deletion and has absolute Jacobian one. It matches dimensions: the smaller-model coefficients and variance have dimension $k+2$, and the one auxiliary draw supplies the larger-model dimension $k+3$. If the reverse death is selected with probability $d_{k+1}$, accept withFor a death, delete the last coefficient $u$, retain the others and the variance, and useTo verify detailed balance, multiply the birth acceptance by $t_k{b}_k{g}_k$ and the reverse death acceptance by $t_{k+1}{d}_{k+1}$. Both equal the minimum of these two quantities in the matched coordinates. This proves reversibility without merely naming a dimension-changing algorithm. A more general bijection introduces its absolute Jacobian, and random choices among several covariates introduce the corresponding forward and reverse selection probabilities. Add within-model updates so that coefficients and variance can explore each model. These are reversible-jump Markov chain Monte Carlo updates; the unspecified model prior and proposal density must be supplied to implement them numerically.