 Solution

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.