# Is multinomial naive Bayes classification not naive Bayes classification?

Suppose I am thinking about a classification problem and I have my features $$X = (X_1,...,X_n)$$ and my classification $$C$$ (taking values in some finite set of classes $$c_1,...,c_k$$). The naive Bayes classifier assumes that the different individual features $$X_1,...,X_n$$ are independent when conditioned on $$C$$. We then construct a classifier by taking the Baysian classifier $$c(x) = \arg\max_{c_i} P(C=c_i | X = x)$$ and by Bayes rule together with our independence assumption, we have $$P(C = c_i | X = x) = \frac{P(C = c_i) \prod_j{P(X_j=x_j | C = c_i)}}{P(X = x)}$$

To get a classifier out of this, we need to be able to compute these probabilities. One way to do this is to make distributional assumptions about the conditional distributions of the $$X_i$$ (for example assuming that they are Bernoulli or Gaussian and then using the data to estimate the parameters of these distributions as well as the marginal distribution of $$C$$).

On the wikipedia page for Naive Bayes it brings up what is called multinomial naive Bayes classification where we assume that the features $$X$$ follow multinomial distributions when conditioned on $$C$$. To get a classifier from this, the parameters of these multinomial distributions are estimated from the data (where we actually do not need the "number of rolls" parameter $$n$$ as it comes out in the wash when taking the argmax) and then used to compute $$P(X = x | C = c_i)$$ (without splitting it up).

One thing to note with a random vector that follows a multinomial distribution is that the different components are not independent (e.g., if you roll a 2 on a dice, you know you do not roll a 5). So the multinomial "naive Bayes" classifier does not satisfy the "naive" assumption that the components of $$X$$ are independent when conditioned on $$C$$.

Am I understanding this correctly? Why multinomial naive Bayes a naive Bayes classifier (my guess, assuming I am understanding everything else correctly: it is naive (making the distributional assumption) and it uses Bayes' rule).

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