In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to explain the difference between the approaches, the author is using the following example(Page 6): If I understand correctly, there are three different cases of protocols (something related to toxicity, it seems) and there are different studies for each of the three protocols, which has examined the relation of the protocol to a specific cause, which is called something as "AAN".
For example, for one of the three protocols, different studies showing the total number of cases (first) and relation to "AAN" (second) is as following :
3) 272, 48
4) 151, 18
... etc. Each of these number pairs belong to different studies.
Now, the Bayesian model for these studies are given as such:
Given $X_j$ is the AAN frequency of the $j$th study about the protocol $i$, $X_j$ is distributed as $X_j \sim Binomial(n_j,p_i)$. $n_j$ is the total number of incidences for the $j$th study and $p_i$ is the parameter of the distribution we want to infer.
What I did not understand here is, the author says that, in this model $p_i$ can vary from study to study.
In my understanding of the Bayesian approach, here, all of the studies constitute our data, $D=\left( n_1,X_1,n_2,X_2,...,n_K,X_K \right)$ where $K$ is the total number of studies. For $p_i$ we have a prior distribution of $P(p_i)$. So we try to find the posterior distribution $P(p_i|D)$. In my understanding $p_i$ cannot vary from study to study: $p_i$ is first generated from the prior distribution as $p_i \sim P(p_i)$ and then this generated value of $p_i$ is used to generate each $X_j \sim Binomial(n_j,p_i)$. Yes, $p_i$ is not fixed as in the frequentist approach, but it varies over different realizations of all $K$ studies, once a $p_i$ is generated from the prior, then all $K$ studies use the same $p_i$. So, it should not change from a single study to another. So, this slide has confused me at this particular point. Am I right with my thoughts here or did I misunderstand something?
Thanks in advance.