It's well known that language can be modeled by Multinomial distribution and Multiple Bernoulli distribution.
So far I don't see any advantage of Multiple Bernoulli distribution representation over Multinomial representation.
Both models equal computationally, Multinomial considers the number of occurrence of the unigram/bigram/.. tehrefore it's more precise.
What's the case when Multiple Bernoulli is preferable over Multinomial.
I presume you came up with this question after looking at naive Bayes models for text classification so I will answers this questions as such.
Event model
I think the question of which event model to use, is very much domain dependent. In many settings it makes more sense to model the data as a series of binary coin flips, each independent of each other. For instance, this recent work on Big Data, shows a superiority of Bernouilli event models in the context of binary transactional data gather from the internet. As a second example, it turns out that when one wants to model human behaviour, models like the Wallenius distribution are better fit for the job.
As far as I know, the multinomial distribution is simply a better model for document classification tasks because it fits the generation process of the data better if you are using a term-frequency representation for your data. A change of representation would require a re-evaluation of the event model in use and perhaps, there a Bernouilli or Wallenius event model could prove to be better.
Computational remarks
Bernouilli and multinomial are not equally expensive, in a term-frequency representation with a lot of zeros in the matrix, a multinomial model will be substantially faster because it does not need to look at the negative information. They are only equally expensive when dealing with binary data.
