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Consider DPM model: $$ \begin{aligned} X_{i} | \phi_{i} & \sim F\left(x;\phi_{i}\right) \\ \phi_{1}, \phi_{2}, \cdots | & P \stackrel{iid}{\sim} P \\ P & \sim D P(\alpha G_0) \end{aligned} $$
It is well-known that $\{\phi_{i}\}$ is exchangeable and $P$ is a.s. discrete. In this model, is $\{X_{i}\}$ also exchangeable?If not, why some scholars still persue non-exchangeable $\{\phi_{i}\}$ modeled by some new process, and then still model $\{X_{i}\}$ using mixture?
Any help will be appreciated!

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