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At temperature $T>0$, target a probability density function proportional to $\exp (f(\theta )/T)$ on a parameter region where it is normalizable, possibly with a specified proper base measure. A Metropolis–Hastings algorithm proposal $q(\theta ,{\theta }^{\prime })$ has acceptance probabilityFor a symmetric proposal, improving moves are accepted and a loss $\Delta$ is accepted with probability $e^{-\Delta /T}$. Run updates while gradually lowering $T$ toward zero. This simulated annealing scheme initially permits exploration and increasingly favors high objective values. Appropriate cooling and mixing conditions are needed for a general global convergence theorem; a finite arbitrary schedule does not guarantee finding a global maximum.

To maximize a positive likelihood function, take $f=\log L$, which has the same maximizers. The resulting likelihood-power annealing target is $L^{1/T}$, and available Gibbs sampler conditional distributions can replace Metropolis proposals.

For minimizing an energy $F(\theta )$, simulated annealing at temperature $T>0$ uses the Boltzmann distribution ${\pi }_T(\theta )\propto e^{-F(\theta )/T}$. At each temperature run a Metropolis–Hastings algorithm or Gibbs transition preserving that law, then reduce $T$. With a symmetric proposal, accept ${\theta }^{\prime }$ with probability $\min \{1,e^{-[F({\theta }^{\prime })-F(\theta )]/T}\}$: downhill moves are accepted and uphill moves remain possible. Keep the best state encountered. As temperature decreases, equilibrium mass favors global minima, provided the target is proper and the relevant concentration hypotheses hold. An arbitrary finite cooling run does not by itself prove global convergence.

Let $s={\sum }_{i=1}^m{x}_i$, $\bar{x}=s/m$ and $S={\sum }_i(x_i-\bar{x}{)}^2$. For the Poisson distribution model the observations must be nonnegative integers. Its log-likelihood isFor $s>0$, differentiation gives ${\ell }_P^{\prime }=-m+s/\lambda$ and ${\ell }_P^{\prime \prime }=-s/{\lambda }^2<0$. If $s=0$, the likelihood is decreasing and the maximizer is the boundary value zero. For the normal distribution model write $v={\sigma }^2>0$, givingFor each $v$, this is maximized at $\mu =\bar{x}$, and differentiating there gives $v=S/m$. HenceThe normal variance estimate has denominator $m$, not $m-1$. A finite positive normal maximum requires $S>0$; if all observations coincide, the likelihood is unbounded as $\mu =\bar{x},v\downarrow 0$. The normal annealing calculation below assumes $S>0$.

To maximize likelihood, choose the energy $-\ell$, so the annealing law is $L^{1/T}$, not $L^{-1/T}$. This sign is important in the Poisson request: literally inserting a positive log-likelihood into a minimizing Boltzmann law would give a nonnormalizable probability density function growing like $e^{m\lambda /T}$ at infinity. We use likelihood-power annealing with the maximizing sign.

For the normal distribution model, take base measure $d\mu dv$. The joint annealing probability density function isCompleting the square gives the Gibbs sampling conditional for $\mu$. Matching the conditional $v$ probability density function to the supplied inverse-gamma distribution convention givesIntegrating out $\mu$ contributes a factor $\sqrt{v}$, so the marginal law isThus this joint Boltzmann law is proper for $T<m/3$. Its variance marginal and the normal conditional both concentrate at the stated estimates as $T\downarrow 0$. For example, for $T<m/7$ its marginal mean and variance areAlso ${\mathbb{E}}_T(\mu -\bar{x}{)}^2=(T/m){\mathbb{E}}_{T}v\to 0$.

There is a direct convergence proof for the actual changing-temperature Gibbs algorithm, beyond convergence of its equilibrium probability densities. Start at a finite deterministic $v_0>0$, choose deterministic $0<T_n\le m/10$ with $T_n\to 0$, draw ${\mu }_n$ conditional on $v_{n-1}$, then draw $v_n$ conditional on ${\mu }_n$ using the two boxed distributions. Conditional normal moments giveThe inverse-gamma distribution first and second moments consequently yieldandIn the first recursion the coefficient of the preceding mean is at most $1/6$; in the second the coefficient of the preceding second moment is at most $1/8$. The means therefore stay bounded, and then so do the second moments. Letting $T_n\to 0$ in these formulas gives $\mathbb{E}{v}_n\to S/m$ and $\mathbb{E}{v}_n^2\to (S/m{)}^2$. Together with $\mathbb{E}({\mu }_n-\bar{x}{)}^2=T_n\mathbb{E}{v}_{n-1}/m\to 0$, this provesThis Gaussian likelihood Gibbs annealing convergence uses the explicit conditional structure and does not assume that one Gibbs sampling sweep is an exact equilibrium draw.

For the Poisson distribution model, the likelihood-power law with respect to $d\lambda$ has the normalized probability density functionsoThe mean tends to $\bar{x}$ and the variance to zero, proving convergence to a point mass at the Poisson maximum-likelihood estimate, including the zero-boundary case.

The Akaike information criterion is $2k-2\ell (\hat{\theta })$, counting one parameter for Poisson and two for normal. With $0\log 0=0$ and $S>0$, the two forms areandChoose the smaller criterion. Likelihood constants that differ between models, including the factorial and $2\pi$ terms, cannot be discarded.

For a fully specified trans-dimensional annealing for penalized likelihood, let $M=P$ or $N$ and defineUse the combined target, relative to counting measure on models and Lebesgue measure on their parameters,where $w_P,w_N>0$ are fixed reference weights. For $S>0$ and $0<T<m/3$ both model components are integrable. The factor $2T$ is a convenient temperature convention keeping the within-model laws exactly those already derived. The extra factor $e^{-k_M/T}$ makes the zero-temperature objective AIC rather than unpenalized likelihood.

Use, for example, probability $1/2$ for a model-change attempt in either model and probability $1/2$ for a within-model update. Within the Poisson distribution model draw a fresh gamma value from its boxed law; within the normal distribution model perform the boxed Gibbs sampling sweep. For a Poisson-to-normal proposal, draw $u\sim q(u)=N(0,{\tau }^2)$ with a fixed ${\tau }^2>0$, and setThe inverse normal-to-Poisson proposal is $\lambda =v$ with reverse auxiliary $u=\mu -v$. This dimension matching uses one auxiliary variable to augment the one-dimensional Poisson parameter space. The absolute Jacobian determinant isIf the forward model-jump selection probability is $b$ and the reverse probability is $d$, the reversible-jump Markov chain Monte Carlo acceptance ratio isAccept with probability $\min (1,R_{P\to N})$. For the reverse move compute $\lambda =v,u=\mu -v$ and useFor the suggested equal selection probabilities, $b=d=1/2$ and those factors cancel. The probability density function $q$ must still be included, despite symmetry of its normal distribution: it is an auxiliary draw in only the forward dimension-increasing proposal. Pairing a move with its inverse makes the accepted forward and reverse probability flows equal after the change of variables, proving detailed balance for each fixed-temperature model-switch kernel.

Cool through proper temperatures, allow enough model and parameter exploration at each stage, and retain the smallest observed $E_M$. The minima of the two energies are exactly the two AIC values above. Low-temperature equilibrium mass concentrates on the better criterion and its fitted parameters; if they tie, the minimizing set contains both models. Unlike the direct normal Gibbs argument, this multimodel exploration claim needs an appropriate cooling/mixing justification and is not guaranteed by an arbitrary short run.

Finally, an exact statistical comparison must put both likelihoods on the same observable and reference-measure convention. A Poisson distribution model is a probability mass function on counts, whereas the displayed normal distribution model is a continuous probability density function. The formulas above are the question's formal Gaussian working-likelihood comparison. A coherent exact normal approximation for count observations can instead use probabilities of rounding intervals; that changes its likelihood and generally loses the simple analytic normal estimates. This distinction matters when interpreting the numerical AIC difference, while leaving the dimension-matching and annealing construction applicable to properly specified likelihoods.