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I have implemented a Naive Bayes Model for a mixture of independent Bernoulli. Where the conditional probability can be written as:

$\mathbb{P}(Y=j | X) \propto \omega_{j} \prod_{i=1}^{d} \mu_{i, j}^{x_{i}}\left(1-\mu_{i, j}\right)^{1-x_{i}}$

Where $\mu_{i, j}=\mathbb{P}\left(x_{i}=1 | Y=j\right)$ and $\omega_{h} :=\mathbb{P}(Y=h)$ where $Y$ is the hidden variable and $x_{i}$ are the observables. The parameters of the model are learned via Expectation-Maximization.

I wanted to generalized this model to non-binary observables by considering a mixture of multinomials. How should I define the conditional probability?

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