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I am reading a paper about the Dirichlet process and image segmentation. (http://dl.acm.org/citation.cfm?id=2404806)

How is (3) below derived? Do any articles or posts explain image segmentation and the Dirichlet process?

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By the Law of Total Probability: $$ p(\boldsymbol X_k^\phi) = \int p(\boldsymbol X_k^\phi, \theta_k) \, \mathrm{d} \theta_k, $$ for which we can rewrite by the definition of conditional probability $$ p(\boldsymbol X_k^\phi, \theta_k) = p(\boldsymbol X_k^\phi \mid \theta_k) \, p(\theta_k), $$ to get the final product $$ \begin{align} p(\boldsymbol X_k^\phi) &= \int p(\boldsymbol X_k^\phi, \theta_k) \, \mathrm{d} \theta_k \\ &= \int p(\boldsymbol X_k^\phi \mid \theta_k) \, p(\theta_k) \, \mathrm{d} \theta_k \\ &= \int \left\{ \prod_{x_n \in \boldsymbol X_k^\phi} p(x_n \mid \theta_k) \right\} \, p(\theta_k) \, \mathrm{d} \theta_k. \end{align} $$ Observe that because we believe the points $x_n$ are assigned clusters independently, we can rewrite $$ p(\boldsymbol X_k^\phi \mid \theta_k) = \prod_{x_n \in \boldsymbol X_k^\phi} p(x_n \mid \theta_k). $$

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