= Solution
Use the <gamma distribution> shape-rate convention: $\Theta\sim\operatorname{Gamma}(\alpha,\beta)$ with $\alpha,\beta>0$. The conditional <Poisson distribution> has mean and variance both equal to $\Theta$, so the <Bühlmann–Straub model> parameters are
$$
\boxed{m_0=v=\frac{\alpha}{\beta},\qquad
a=\frac{\alpha}{\beta^2},\qquad \frac va=\beta.}
$$
For $W=\sum_jm_j$ and $y=\sum_jy_j$, the <Bühlmann–Straub credibility factor> is $Z=W/(W+\beta)$. Thus
$$
\widehat\mu=\frac{W}{W+\beta}\frac{y}{W}
+\frac{\beta}{W+\beta}\frac{\alpha}{\beta}
=\frac{y+\alpha}{W+\beta},
$$
and the required expected-count estimate is
$$
\boxed{\widehat{\mathbb E[Y_{n+1}\mid\Theta]}
=m_{n+1}\frac{\alpha+\sum_{j=1}^n y_j}{\beta+\sum_{j=1}^n m_j}.}
$$
The total exposure determines how much information the observed count carries; the number of years alone is not the appropriate denominator.
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