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I want to use the EM algorithm to do clustering under a missing labels regime. The assumption I am making about the missing data is that it's distributed according to Bernoulli distribution.

So for the complete data likelihood we have:
$$ P_{X,Z}(\mathcal{D}, Z; \theta) = \prod_{i=1}^{N}P_{X,Z}(x_i,z_i; \theta) = \prod_{i=1}^{N}[P(x_i|z_i, \theta)P(z_i)]^{z_{i}} $$

My question is this: is there a general guideline for making assumptions on the class priors $P(z_i)$ ?

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You can use any prior you feel is reasonable, based on your prior knowledge, however if you want to be able to solve the equations in closed-form, you would want to choose a congruent prior, which means it takes the same form as the conditional distribution of $x$ with respect to $z$.

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