# Understanding Exchangability in Bayesian Hierarchical Models

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!!!