# Multinomial Distribution for Naive Bayes

I'm mainly confused because of some of the wording in this CV post. Multinomial Naive Bayes.

Mainly, this line:

In summary, Naive Bayes classifier is a general term which refers to conditional independence of each of the features in the model, while Multinomial Naive Bayes classifier is a specific instance of a Naive Bayes classifier which uses a multinomial distribution for each of the features." by jlund3.

Why is each feature $x_i$ a multinomial distribution, and not the product $\prod{P(x_i|c)}$ a multinomial distribution?

$$P(c|X) \propto \prod{P(x_i|c)} * P(c)$$

If the product of probabilities is distributed as a multinomial, that makes sense to me since

$$\prod{P(x_i|c)} \propto \prod{{p_i}^{x_i}}$$

I don't really understand how each feature itself could be distributed as a multinomial, however. Wouldn't each $x_i$ end up being a multinomial with two labels ($x_i$ == count of x_i. $x_{not i}$ == count of every other word)

Any help would be much appreciated!

Take throwing a dice for example, the result $$X$$ can be a random number in $$1,2,..,6$$. When we do Bayesian parameter estimation, we can assume the probability of throwing a $$1$$ is $$p_1$$, throwing a $$2$$ is $$p_2$$,..., and so on. Hence, we want to estimate $$(p_1,p_2,...,p_6)$$. If we throw the dice $$N$$ times, we get $$1$$ for $$c_1$$ times, $$2$$ for $$c_2$$ times. We denote the result as a vector $$\textbf{c}=(c_1,c_2,..,c_6)$$, where $$c_1+c_2+...+c_6=N$$. Then the likelihood will be $$\prod_{j=1}^6p_{j}^{c_j}$$ ,which is exactly multinomial distribution. If $$N=1$$, only one dimension of $$\textbf{c}$$ will be 1, the rest will be 0. As you can see, this feature $$X$$ is a multinomial distribution.