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In subsection 25.2.1 it's stated, regarding finite mixture model:

The usual representation (of a finite mixture model) is as follows:

$p(x_i|z_i = k, \boldsymbol\theta) = p(x_i|\boldsymbol\theta_k)$

$p(z_i = k| \boldsymbol\pi = \pi_k) = \pi_k$

$p(\boldsymbol\pi|\alpha) =\text{Dir}(\boldsymbol\pi|(\alpha/K)\boldsymbol1_K)$

The form of $p(\boldsymbol\theta_k|\lambda)$ is chosen to be be conjugate to $p(x_i|\boldsymbol\theta_k)$. We can write $p(x_i|\boldsymbol\theta_k)$ as $\boldsymbol{x}_i \sim F(\boldsymbol\theta_{z_i})$ where F is the observation distribution. Similarly, we can write $\boldsymbol\theta_k \sim H(\lambda)$, where H is the prior.

I don't agree on

We can write $p(x_i|\boldsymbol\theta_k)$ as $\boldsymbol{x}_i \sim F(\boldsymbol\theta_{z_i})$ where F is the observation distribution

Because $F(\boldsymbol\theta_{z_i})$ is the conditional probability of x given z. So in my opinion it should be:

We can write $p(x_i|\boldsymbol\theta_k)$ as $\boldsymbol{x}_i|z_i \sim F(\boldsymbol\theta_{z_i})$ where F is the observation distribution

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  • $\begingroup$ Correct, this is the conditional distribution. $\endgroup$ – Xi'an Jan 11 at 13:29

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