# Topic-prevalence in Latent Dirichlet Allocation

In Focused Topic Models, the main motivation is to decouple the global topic prevalence and in-document prevalence. However, I couldn't see how the original Latent Dirichlet Allocation (LDA) couples those two prevalences. In LDA, if we use a very low concentration parameter ($$\alpha$$) to enable sparsity, then we can almost be sure that once we select an exposure $$\theta$$ (probability distributions over topics), then the document will be dominated by very few topics, and this topic could be a rare topic globally, but it will dominant in the document.

The only thing I can think about is that the above occasion is actually very rare, since we have no way to control $$\theta$$ in LDA, so at the end of the day, what we have are documents dominated by prevalent topics, even some document should be better generated by rare topics. Therefore, introducing the binary matrix can soothe the situation.