Trans-dimensional annealing for penalized likelihood (${\Pi }_T(j,\theta )\propto w_j{e}^{-{\mathrm{AIC}}_j(\theta )/(2T)}$)

To minimize $E_j(\theta )=2k_j-2{\ell }_j(\theta )$ over models with different parameter dimensions, sample from a target proportional to $w_j{e}^{-E_j(\theta )/(2T)}$ and cool $T$. A dimension-matching proposal augments the smaller model by auxiliary variables, transforms them bijectively into the larger parameter vector, and includes the proposal probability density function and absolute Jacobian determinant in the Metropolis–Hastings acceptance probability. For a positive Poisson mean $\lambda$ and normal parameters $(\mu ,v)$, one convenient transformation isIts absolute Jacobian determinant is one. If the forward and reverse model-jump selection probabilities are $b$ and $d$, the Poisson-to-normal ratio isThe reverse ratio is its reciprocal at the inverse map. This enforces detailed balance. When the target is proper and the annealing chain explores it adequately, the low-temperature laws concentrate on the minimum Akaike information criterion, rather than the maximum marginal model evidence. Likelihoods compared across models must describe the same observed data with a compatible reference measure.

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.