= Solution
In the <Bühlmann model>, a latent risk parameter $\Theta$ is drawn from a population distribution. Conditional on $\Theta$, the yearly observations are <independent and identically distributed random variables>, with <conditional expectation> $m(\Theta)$ and <conditional variance> $v(\Theta)$. Define the structural parameters
$$
m=\mathbb E[m(\Theta)],\qquad
v=\mathbb E[v(\Theta)],\qquad
a=\operatorname{Var}(m(\Theta)).
$$
Here $v$ is the <expected process variance>, while $a$ is the <variance of hypothetical means>. The <Bühlmann credibility premium> is the best affine estimate of $m(\Theta)$ from the observed claims, under <mean squared error>. Predicting the next claim gives the same affine estimate: the extra conditional observation noise contributes the constant $v$ to the prediction error.
The <law of total variance> and <conditional independence> give
$$
\operatorname{Var}(X_j)=a+v,\qquad
\operatorname{Cov}(X_i,X_j)=a\quad(i\ne j),\qquad
\operatorname{Cov}(m(\Theta),X_j)=a.
$$
An affine estimate can be written as $\widehat m=m+\sum_{j=1}^n b_j(X_j-m)$: for any chosen $b_j$, optimizing the constant makes its <expected value> equal to $m$. The normal equations for the <linear least-squares projection> are
$$
v b_i+a\sum_{j=1}^n b_j=a,\qquad i=1,\ldots,n.
$$
For $v>0$ they force all $b_i$ to agree, with $b_i=a/(v+na)$. Thus \b[the credibility factor and premium are]
$$
\boxed{Z=\frac{na}{v+na}=\frac{n}{n+v/a},\qquad
\widehat m=Z\overline X+(1-Z)m,}
\qquad \overline X=\frac1n\sum_{j=1}^nX_j.
$$
The <credibility factor> increases with the observation count and between-risk <variance>, and decreases with within-risk <variance>. If $a=0$, the risk mean is known and $Z=0$; if $v=0$ and $a>0$, one observation reveals it and $Z=1$. If both vanish, the premium is the fixed value $m$ and the factor is immaterial.
In the specified model, the conditional law is a <gamma distribution> with shape $\alpha$ and scale $\theta$. Therefore
$$
m(\theta)=\alpha\theta,\qquad v(\theta)=\alpha\theta^2.
$$
The prior is an <inverse-gamma distribution> with shape $k$ and scale $\lambda$. To obtain its moments directly, substitute $y=\lambda/\theta$ in the defining integral, obtaining
$$
\mathbb E[\Theta^r]
=\lambda^r\frac{\Gamma(k-r)}{\Gamma(k)},\qquad r<k.
$$
The <Gamma function recurrence> yields
$$
\mathbb E\Theta=\frac{\lambda}{k-1},\qquad
\mathbb E\Theta^2=\frac{\lambda^2}{(k-1)(k-2)},\qquad
\operatorname{Var}(\Theta)=\frac{\lambda^2}{(k-1)^2(k-2)}.
$$
The assumption $k>2$ makes both structural <variances> finite. Hence
$$
m=\frac{\alpha\lambda}{k-1},\qquad
v=\frac{\alpha\lambda^2}{(k-1)(k-2)},\qquad
a=\frac{\alpha^2\lambda^2}{(k-1)^2(k-2)},\qquad
\frac va=\frac{k-1}{\alpha}.
$$
It follows that \b[the model-specific credibility estimate is]
$$
\boxed{Z=\frac{n\alpha}{n\alpha+k-1},\qquad
\widehat m_{\rm B}
=Z\overline X+(1-Z)\frac{\alpha\lambda}{k-1}
=\frac{\alpha(\lambda+\sum_{j=1}^nX_j)}{k+n\alpha-1}.}
$$
For the <Bayes estimator under squared error loss>, the quantity to estimate is $\alpha\Theta$, so the optimum is its <posterior mean>. The <likelihood function>, viewed as a function of $\theta$, is proportional to
$$
\theta^{-n\alpha}\exp\left(-\frac{\sum_jx_j}{\theta}\right).
$$
Multiplication by the prior shows <gamma scale inverse-gamma conjugacy>:
$$
\Theta\mid x_1,\ldots,x_n
\sim\operatorname{InvGamma}\left(k+n\alpha,\lambda+\sum_jx_j\right).
$$
Although the printed hint only mentions integer shapes, the same substitution and <Gamma integral> normalize this posterior for every positive real shape, so no integrality of $\alpha$ is needed. Its <posterior mean> gives \b[the Bayesian estimate and comparison]
$$
\boxed{\widehat m_{\rm Bayes}
=\alpha\mathbb E[\Theta\mid X_1,\ldots,X_n]
=\frac{\alpha(\lambda+\sum_{j=1}^nX_j)}{k+n\alpha-1}
=\widehat m_{\rm B}.}
$$
This <exact Bühlmann credibility for gamma claims> holds for every observed sample, not merely on average. Here the <posterior mean> is affine in the <sample mean>, so the best affine <Bühlmann credibility premium> is also the unrestricted <Bayes estimator under squared error loss>.
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