Assume that $N_t | \lambda \sim \textrm{Pois}(\lambda t)$ and $\lambda \sim \textrm{Gamma}(\alpha, \beta)$. It can be shown that $N_t$ is Negative Binomial with $r = \alpha$ and $p=\frac{t}{t+\beta}$.
Now lets assume that we want to know the value of $N_t$ at some time $T$. We have observed the process in the interval $[0,s]$ for $s < T$ and want to draw inference on $\lambda$. How would this be done?
I was thinking, that if $\lambda$ had a Gamma distribution before-hand, surely the information of $N_s = n$ would allow to us to infer something about the likely values of $\lambda$? I'm interested in both frequentist and bayesian approaches.