= Trans-dimensional annealing for penalized likelihood
{title2=$\Pi_T(j,\theta)\propto w_j e^{-\mathrm{AIC}_j(\theta)/(2T)}$}
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 is
$$
u\sim N(0,\tau^2),\qquad(\mu,v)=(\lambda+u,\lambda),
\qquad(\lambda,u)=(v,\mu-v).
$$
Its absolute <Jacobian determinant> is one. If the forward and reverse model-jump selection <probabilities> are $b$ and $d$, the Poisson-to-normal ratio is
$$
R=\frac{w_Nd}{w_Pb\,q(u)}
\exp\!\left[-\frac{E_N(\lambda+u,\lambda)-E_P(\lambda)}{2T}\right].
$$
The 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.
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