= Birth and death moves for Bayesian variable selection
{title2=$\alpha_{k\to k+1}=\min\!\left(1,\frac{t_{k+1}(a,u,v)d_{k+1}}{t_k(a,v)b_k g_k(u\mid a,v)}\right)$}
For nested regression models, append one coefficient $u$ drawn from density $g_k$ while retaining the variance $v$ and existing coefficients $a$. The dimension-matching map $(a,v,u)\mapsto(a,u,v)$ has absolute Jacobian one. With joint model/parameter target $t_k$, birth-selection probability $b_k$, and reverse death-selection probability $d_{k+1}$, the displayed acceptance probability and its reciprocal death rule enforce <detailed balance>. Include normalized parameter-prior densities and a model-order prior in $t_k$: constants that depend on model order cannot be dropped from a cross-model ratio.
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