= Binomial-normal reversible-jump annealing
{c}
{title2=$(p,u)\mapsto(\mu,v)=(\log[p/(1-p)],e^u)$}
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 ratio
$$
R=\exp\!\left(\frac{\ell_N-2-\ell_B+1}{T}\right)\frac d{bh(u)}\frac v{p(1-p)}.
$$
The 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.
Back to article page
