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Let $s={\sum }_{j=1}^n{x}_j$. The conditional joint likelihood function of the observed counts isThe prior is a gamma distribution with shape two and rate $\lambda$. Multiplication by its density gives the unnormalized posterior density ${\theta }^{s+1}{e}^{-(n+\lambda )\theta }$. Normalizing with the gamma function yieldsThus Poisson-gamma conjugacy gives $\Theta \mid \mathbf{x}\sim \mathrm{Gamma}(s+2,n+\lambda )$. Conditional on $\Theta$ and the past data, the next count has mean $\Theta$, so the law of total expectation givesWith $\overline{x}=s/n$ and prior mean $m_0=2/\lambda$, this is exactlyHere $n\ge 1$, $\lambda >0$, and the credibility factor increases to one as the number of observed years grows. The denominator uses the prior rate $\lambda$ and total exposure $n$; it does not add two to the exposure when adding the prior shape.

A credibility estimate combines a risk's observed experience with collective or prior information, typically as $Z\overline{X}+(1-Z)m_0$. The credibility factor $Z\in [0,1]$ is the weight assigned to the individual experience: more reliable or more extensive data generally increases it. Bayesian credibility treats the risk parameter as random with a prior distribution and predicts future claims by averaging the conditional claim mean under its posterior distribution. Under squared-error loss this posterior predictive mean is optimal. It need not be an affine combination in every Bayesian model; the Poisson-gamma conjugacy calculation below yields that credibility form exactly, rather than as a best-linear approximation.

A proper gamma approximation to a Poisson Jeffreys prior uses shape $1/2$ and a small positive rate $\epsilon$:Its density is proportional to ${\lambda }_i^{-1/2}{e}^{-\epsilon {\lambda }_i}$, so where $\epsilon {\lambda }_i$ is small its kernel approximates the Jeffreys prior. After observing the data, Poisson-gamma conjugacy gives $\mathrm{Gamma}(y_i+1/2,E_i+\epsilon )$, which converges to the preceding posterior distribution as $\epsilon \downarrow 0$. This is an approximation of an improper kernel on the relevant parameter range, not a weak limit to a normalized Jeffreys probability distribution. The shape must be $1/2$; a small-shape gamma prior would approximate a different power of ${\lambda }_i$.

The reversible-jump Markov chain Monte Carlo state comprises the model index and that model's parameter. Its unnormalized posterior density in model $j$ isFor models of equal dimension, choose model $k\ne j$ with probability $s_{jk}(\theta )$ and propose ${\theta }^{\prime }$ using density $q_{jk}({\theta }^{\prime }\mid \theta )$ on the destination parameter space. The equal-dimension reversible-jump acceptance probability isOn acceptance change both model and parameter; on rejection retain both. Choose model proposals connecting all models of positive posterior mass and within-model kernels exploring their supports, with aperiodicity, to obtain an ergodic chain. Include within-model updates targeting its conditional posterior, and use model occupation proportions after the initial transient to estimate posterior model probabilities. The formula includes the model prior probabilities, parameter prior densities, likelihoods, reverse model-selection probability, and reverse parameter-proposal density.

This direct-density formula is Metropolis–Hastings algorithm on a disjoint union of equal-dimensional spaces. If instead a move uses a deterministic bijection $T:(\theta ,u)\mapsto ({\theta }^{\prime },u^{\prime })$ with auxiliary proposal densities $g_{jk}(u\mid \theta )$ and $g_{kj}(u^{\prime }\mid {\theta }^{\prime })$, useThe Jacobian determinant is one for identity matching, but equal model dimensions alone do not make a nonlinear map volume-preserving. For a direct proposal density the change of variables is already included in that density, so no additional Jacobian factor is inserted.

For the two Poisson models, use the shape-rate convention for the gamma distribution:If $b$ denotes scale instead, replace every occurrence of the rate $b$ below by $1/b$. Introduce the inactive ${\gamma }_2$ under model $2$ with the proper pseudo-prior $\pi ({\gamma }_2)$, independent of ${\gamma }_1$. This pseudo-prior augmentation for model comparison does not change model $2$'s marginal likelihood because the inactive density integrates to one. The two augmented targets on the same positive quadrant arewhereUse a symmetric cross-model proposal that flips the model label and leaves both parameters unchanged. The matching map is the identity, with unit Jacobian determinant, and the common prior factors cancel. Hence the model-switch acceptance probabilities areOnly the second observation's factor changes because ${\gamma }_1$ is retained as its shared candidate mean. Evaluate the ratio on a logarithmic scale as ${\gamma }_2-{\gamma }_1+x_2(\log {\gamma }_1-\log {\gamma }_2)$ for the first direction.

For within-model moves, Poisson-gamma conjugacy gives exact Gibbs sampling refreshes:The two draws are conditionally independent within either model. Choose with fixed positive probabilities between this block refresh and the cross-model proposal. Both moves preserve the augmented posterior; refreshing the inactive parameter maintains its correct distribution and supports mixing. The fraction of retained states labelled $M_j$ estimates $\mathbb{P}(M_j\mid x)$.

An equivalent literal dimension-changing implementation deletes ${\gamma }_2$ on a proposed $1\to 2$ move, and on a $2\to 1$ move draws an auxiliary $u\sim \mathrm{Gamma}(a,b)$ and sets ${\gamma }_2=u$. The birth matching has $1+1=2+0$ dimensions and unit Jacobian determinant. Its proposal density cancels the new parameter's prior density, giving the same acceptance ratios above for symmetric selection of move directions. This provides suitable moves without storing an inactive parameter.

For an independent check of Poisson mean equality model comparison, the exact Bayes factor isIt follows by integrating the two gamma likelihood kernels; the common factorial terms cancel. With equal model priors, the exact posterior probability of $M_2$ is $m_2/(m_1+m_2)$, which can also be used to check the RJ-MCMC occupation estimate.

By Poisson-gamma conjugacy, the posterior density is proportional toThus the shape-rate posterior isIts posterior mean is $A/B$ and its variance is $A/B^2$. The parameter $B$ is a rate, rather than a scale.

Use the shape-rate convention $R_k\sim \mathrm{Gamma}({\alpha }_0,{\beta }_0)$. The factors involving $R_k$ in the posterior density areBy Poisson-gamma conjugacy,so its posterior mean is $(I_k+{\alpha }_0)/(s_k+{\beta }_0)$.

For an observed value $X=x$, the likelihood iswhile the gamma prior with shape $\alpha$ and rate $\lambda$ has densityThe posterior density is therefore proportional toNormalizing this gamma kernel proves the Poisson-gamma conjugacy formula

Bayes' theorem updates a hypothesis using observed data. For mutually exclusive exhaustive hypotheses $H_j$ and data $D$ of nonzero probability,The prior $P(H_j)$ describes information before these data; the likelihood $P(D\mid H_j)$ describes the sampling model; the denominator is the evidence; and the posterior is the updated probability. likelihood and posterior are different conditional quantities. For a continuous parameter, sums become integrals and the posterior is a probability density.

Consider a counter registering independent events at unknown constant rate $\lambda$, observed for known exposure $T$ with negligible background and dead time. The observed count has a Poisson distribution:Take a proper gamma prior in shape-rate convention,Multiplying by the likelihood adds the count to the shape and the exposure to the rate. Normalization therefore givesHere $\beta$ has exposure units. This is Poisson-gamma conjugacy. Successive independent counting runs update the same posterior by adding their counts and exposures, so the result agrees with analysing all the data together.

A credible interval contains a specified posterior probability for $\lambda$, conditional on the chosen prior and model. For low counts, the posterior can be asymmetric and sensitive to the prior, and a symmetric Gaussian error bar can be misleading. For large $n$ and an exposure dominating prior information, the mean approaches $n/T$ and its standard deviation approaches $\sqrt{n}/T$. Background, missed events or time-dependent rates require a richer likelihood; Bayes' theorem cannot compensate for an incorrect physical counting model. The prior must be specified independently of the same data being used in the likelihood, unless its estimation is explicitly included in a larger model.

Bayes' theorem updates prior knowledge using the likelihood of new observations. For alternative hypotheses $H_j$,The prior $P(H_j)$ describes information before the data $D$; $P(D\mid H_j)$ is the sampling likelihood, and the posterior $P(H_j\mid D)$ describes the updated probabilities. Posterior odds equal prior odds times the likelihood ratio. A likelihood is a probability model for data at specified parameters, not automatically a normalized probability distribution over those parameters.

For a continuous parameter $\theta$, the same rule uses densities: $\pi (\theta \mid D)=L(D\mid \theta )\pi (\theta )/\int L(D\mid u)\pi (u)du$. Priors can encode positivity, physical limits, previous independent measurements or calibration knowledge. Their parameterization matters: a density uniform in a rate is not uniform in its logarithm. Proper priors make model evidences well defined; an arbitrary normalization in an improper prior is especially problematic for comparing models.

A counting example uses Poisson-gamma conjugacy. If a detector registers $K$ events in exposure $t$ with constant rate $\lambda$, $L\propto {\lambda }^K{e}^{-t\lambda }$. A gamma prior with shape $\alpha$ and rate parameter $\beta$, $\pi (\lambda )\propto {\lambda }^{\alpha -1}{e}^{-\beta \lambda }$, givesThe posterior mean combines the prior rate $\alpha /\beta$ with the observed rate $K/t$, weighted by the prior and actual exposures. The parameters must use a consistent shape-rate convention. Small counts can leave substantial prior sensitivity; abundant informative data usually reduce it for regular identifiable models.

Independent old data can become the prior for a new experiment, giving the same result as joint updating under a consistent model. The new data must not also have been used to construct that prior, or the evidence is counted twice. A credible interval has a specified posterior probability under the chosen model and prior; this interpretation differs from repeated-sampling coverage of a confidence interval. Predictions for a future observation average its sampling distribution over the posterior, incorporating parameter uncertainty. Prior sensitivity checks and a defensible likelihood are therefore part of the inference, rather than Bayes' theorem itself supplying missing physical knowledge.

Writing $s={\sum }_i{x}_i$, the Poisson distribution likelihood is proportional to ${\theta }^s{e}^{-n\theta }$. Multiplying by the gamma distribution prior givesThis is Poisson-gamma conjugacy, with the rate convention made explicit:Its normalized density is $(\lambda +n{)}^{r+s}{\theta }^{r+s-1}{e}^{-(\lambda +n)\theta }/\Gamma (r+s)$ for $\theta >0$.

Compare independent Poisson means with one shared mean under proper shape-rate gamma distribution priors. Adding the second mean as an inactive parameter under the shared-mean model gives a simple identity model switch. With equal model priors and identical active/inactive priors, its ratio from separate to shared means is $e^{{\gamma }_2-{\gamma }_1}({\gamma }_1/{\gamma }_2{)}^{x_2}$. Poisson-gamma conjugacy supplies exact within-model updates, and integrating the gamma kernels supplies an independent Bayes factor benchmark.