= Solution
The specified <gamma distribution> uses shape $k$ and rate $\lambda$. Multiplying its density by the Poisson likelihood gives a posterior density proportional to $\theta^{k+S-1}e^{-(\lambda+n)\theta}$. Thus the <conjugate prior> calculation gives
$$
\boxed{\theta\mid X_1,\ldots,X_n\sim\Gamma(k+S,\lambda+n).}
$$
For <squared-error loss>, the posterior expected loss is
$$
\mathbb E[(\theta-a)^2\mid X]=\operatorname{Var}(\theta\mid X)
+(a-\mathbb E[\theta\mid X])^2.
$$
It is minimized at the <posterior mean>, so the <Bayes estimator> is
$$
\boxed{\widehat\theta_{\rm B}=\frac{k+S}{\lambda+n}.}
$$
For a fresh conditionally independent Poisson observation, the <posterior predictive probability> of zero is the posterior expectation of $e^{-\theta}$. Evaluating the gamma integral gives
$$
\boxed{\mathbb P(X_{n+1}=0\mid X_1,\ldots,X_n)
=\left(\frac{\lambda+n}{\lambda+n+1}\right)^{k+S}.}
$$
The samples are independent conditional on $\theta$; integrating out the uncertain common parameter produces the predictive probability above.
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