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I have a group of $k$ experiments (indexed by $j=1...k$) and each experiment $j$ produces a set of $n_j$ datapoints denoted as $y_{ij}$ such that $i\in\{1,...,n_j\}$. Meanwhile, $y_{ij}$ is distributed according to an experiment-related parameter $\theta_j$ and $\theta_j$ is generated according to a distribution with a common parameter $\phi$: $$ y_{ij} \sim Dist_1(\theta_j)\\ \theta_j\sim Dist_2(\phi) $$

I am trying to get the posterior $P(\phi|y)$ where $y= \{y_{ij}\} \forall i,j$. When computing the posterior, it is necessary to marginalize over all $\theta_j$ for $P(\phi,\theta_1,\theta_2...|y)$, however I am not sure:

  1. Are $\theta_j$s in my case exchangable?
  2. If the marginalization is over all permutations of $\theta_j$s? That is, all possible values of the vector $\theta=[\theta_1,...,\theta_k]$.

Any help is much appreciated!!!

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