= Solution
Use the shape-rate convention for the <gamma distribution> $\theta\sim\operatorname{Gamma}(\alpha,\beta)$, with <prior distribution> <probability density function> proportional to $\theta^{\alpha-1}e^{-\beta\theta}$. The observations are <independent> conditional on the common parameter $\theta$. Unconditionally a common nondegenerate <prior distribution> induces <covariance> $\alpha/\beta^2$ between distinct observations, so the <Bayesian inference> model requires conditional <independence>.
For unit-exposure counts, the <likelihood> is proportional to $\theta^{\sum_i x_i}e^{-n\theta}$. Multiplying by the <prior distribution> gives
$$
\theta\mid\mathbf x\sim\operatorname{Gamma}\left(\alpha+\sum_{i=1}^n x_i,\ \beta+n\right).
$$
The future count has <conditional expectation> $\theta$, so <posterior distribution> averaging gives
$$
\boxed{\mathbb E[X_{n+1}\mid\mathbf x]
=\frac{\alpha+\sum_i x_i}{\beta+n}
=\frac{n}{n+\beta}\overline x+\frac{\beta}{n+\beta}\frac\alpha\beta.}
$$
This is an exact <Bayesian credibility> estimate, with <credibility factor> $Z=n/(n+\beta)$ and collective <mean> $\alpha/\beta$.
For the <insurance exposure> and <claims inflation> calculation put $c_j=c(1+r)^j$, assuming $c>0$, $1+r>0$ and positive policy counts $n_j$. Let $K_j$ be the total claim count in year $j$. <Conditional independence> of the policy counts gives $K_j\mid\theta\sim\operatorname{Pois}(n_j\theta)$, and the observed average amount satisfies
$$
Y_j=\frac{c_jK_j}{n_j},\qquad k_j=\frac{n_jy_j}{c_j}.
$$
For possible observations, the recovered $k_j$ are nonnegative integers. If they are not, the data have zero <likelihood> under the model and no conditional <posterior distribution> is defined for those impossible values.
Write $W=\sum_{j=1}^n n_j$ and $K=\sum_{j=1}^n k_j$. Since $Y_j$ and $K_j$ determine each other, conditioning on the observed average amounts is equivalent to conditioning on these counts. The <independent> yearly count <likelihood> is
$$
\prod_{j=1}^n e^{-n_j\theta}\frac{(n_j\theta)^{k_j}}{k_j!}
\propto\theta^K e^{-W\theta}.
$$
The <Poisson-gamma conjugacy with unequal exposures> therefore gives
$$
\boxed{\theta\mid\mathbf y\sim\operatorname{Gamma}(\alpha+K,\beta+W).}
$$
For the future year, $\mathbb E[Y_{n+1}\mid\theta]=c_{n+1}\theta$; the number of future policies cancels because $Y_{n+1}$ is a per-policy average. Hence
$$
\boxed{\mathbb E[Y_{n+1}\mid\mathbf y]
=c_{n+1}\frac{\alpha+K}{\beta+W}
=Z\,m(\mathbf y)+(1-Z)m,}
$$
with
$$
\boxed{\begin{aligned}
Z&=\frac{W}{W+\beta},\\
m&=c(1+r)^{n+1}\frac\alpha\beta,\\
m(\mathbf y)&=c_{n+1}\frac KW
=\frac1W\sum_{j=1}^n n_j(1+r)^{n+1-j}y_j.
\end{aligned}}
$$
The collective <mean> $m$ is the <prior distribution> expected claim amount per policy in the future year's monetary units. The experience <mean> $m(\mathbf y)$ first inflates each past per-policy amount to that same future year and then averages with policy-exposure weights. Equivalently, it estimates annual claim frequency by total observed claims divided by total past <insurance exposure> and converts that frequency to the future claim cost. This is <inflation-adjusted Poisson-gamma credibility>.
For fixed <prior distribution> parameters, $Z$ increases strictly with total past <insurance exposure>: $dZ/dW=\beta/(W+\beta)^2>0$, and \b[$Z\to1$ as $W\to\infty$]. The <prior distribution> weight decreases to zero; $\beta$ acts as equivalent <prior distribution> <insurance exposure>. With no past <insurance exposure> the appropriate weight is zero and the prediction is $m$, without defining an empirical experience <mean>.
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