Naive Bayes Multinomial, independence assumption misunderstood This is embarrassing but I think I miss understand something.


In multinomial distribution, "while the trials are independent, their outcomes X are dependent because they must be summed to n." wiki
Naive Bayes assumption is that the features are independent (given the class): "assume that each feature $x_{i}$ is conditionally independent of every other feature" wiki


Thus, how can 1&amp;2 co-exist in what's known as the Multinomial Naive-Bayes model?

Specifically, $p(x_i|c)=p_i^{x_i}(1-p_i)^{n-x_i}\binom{n}{k}$ and $p(x_1,...,x_n)=\frac{n!}{\prod_i{x_i}!}\prod{p_i^{x_i}}$ thus $p(x_1,...,x_n) \ne \prod_ip(x_i|c) $
Their outcomes X are dependent because they must be summed to n  


If you are really want to model multiple experiments probabilistically, please do not forget the normalization constant that make the unnormalized quantity normalized by integrating/summing them to one.   

Multinomial Naive Bayes model, as described by Manning et al (2008), estimates the conditional probability of a particular word/term/token given a class as the relative frequency of term t in documents belonging to class c:  



Thus this variation takes into account the number of occurrences of term t in training documents from class c, including multiple occurrences Not multiple experiments.  

see The Naive Bayes Text Classifier