Empirical Bayes estimator for a Beta-Binomial parameters

Question

Let $X_t$ be collected from a Binomial distribution with parameters $N_t$ and $P_t$, where $N_t$ is known for $t= 1, 2, \dots , T$. On the other hand, $P_t \sim \operatorname{Beta}(\alpha_t, \beta_t)$. Using an Empirical Bayes estimation, we are interested in getting an estimate of $\alpha_t$ and $\beta_t$ based on the observations of $x_t$ we have.

Answer 1

I think the question itself shows a lot of confusion. If you want EB estimates of0 $P_t$, you pretty much have to put a common prior on all $P_t$, i.e., $\alpha$ and $\beta$ cannot depend on $t$. If you want $\alpha$ and $\beta$ to depend on $t$, then perhaps you will need another level of prior on $\alpha_t$ and $\beta_t$. Even so, your set up has a potential identifiability problem, but the key is that, eventually, you need a common prior for all $t$.