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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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Since the observables in the original problem are binary you just need one parameter, $\mu_{i,j}$, for each observable conditioning on a given y and the other one can be got by $1-\mu_{i,j}$.

When it comes to multinomial observables that would be K-1 where K is the number of classes of the observation variable. Thus, you can change $\mathbb{P}(Y=j | X) \propto \omega_{j} \prod_{i=1}^{d} \mu_{i, j}^{x_{i}}\left(1-\mu_{i, j}\right)^{1-x_{i}}$ to $\mathbb{P}(Y=j | X) \propto \omega_{j} \prod_{i=1}^{d} \prod_{k=1}^K\mu_{i, j,k}1\{x_i=k\}$.

And $\mu_{i, j,k}=\mathbb{P}\left(x_{i}=k | Y=j\right)$ where $x_i$ can take k values and $k>2$, and $1\{x_i=k\}$ equals 1 if $x_i$ is the kth class and 0 otherwise.

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