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I have a Bayesian hierarchical model with datapoints $y_{ij}$ which are samples from distributions with parameters $\theta_j$. For each distribution parameter $\theta_j$, there are $n_j$ datapoints generated. The distribution parameters themselves are independent samples from another distribution with parameter $\psi$. Both $\theta=\{\theta_1,...,\theta_j\}$ and $\psi$ are unobservable. I am trying to get the posterior distribution of $\psi$ given the datapoints $y = \{y_{ij}\}\forall i,j$.

I started with the basic Bayesian hierarchical model formula: $$ P(\psi|y)\propto P(y|\psi)P(\psi)=P(\psi|y)\propto \sum_{\theta}P(y|\theta)P(\theta|\psi)P(\psi) $$ I understand that both $y$ and $\theta$ are vectors, therefore I have a few questions:

  1. How to compute $P(y|\theta)$ since I only have $P(y_{ij}|\theta_j)$? I can compute $P(y_j|\theta_j)=\prod_i P(y_{ij}|\theta_j)$ but I don't think $P(y|\theta)=\prod_j P(y_j|\theta_j)$, right?
  2. If $\theta$ is a vector of length $n$ (there are $n$ distributions generating datapoints) an state sapce size of each $\theta_j$ is 3, does the total number of available $\theta$ to be marginalized over be $3^n$? I am trying to figure out how many permutations of $\theta$ I need to sum over here, but I realize there is the exchangability condition which I am not sure if it has an effect on this or not.

Any help is deeply appreciated! Thanks!

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