Binomial-normal reversible-jump annealing ($(p,u)\mapsto (\mu ,v)=(\log [p/(1-p)],e^u)$)

For one-parameter binomial and two-parameter normal models, the displayed dimension-matching map has inverse $(p,u)=((1+e^{-\mu }{)}^{-1},\log v)$ and absolute Jacobian determinant $v/[p(1-p)]$. With auxiliary density $h(u)$ and model-jump probabilities $b,d$, an annealing target proportional to $e^{({\ell }_j-k_j)/T}$ gives forward acceptance ratioThe reverse ratio is its reciprocal at the inverse map. Keep model-dependent likelihood constants. Comparing continuous densities directly with discrete masses is not invariant to measurement units; a scientifically meaningful comparison needs models for the same recorded observations with a compatible reference measure, for example bin-integrated normal probabilities for integer-valued measurements.