Why do we need the hyperparameters beta and alpha in LDA?
I'm trying to understand the technical part of Latent Dirichlet Allocation (LDA), but I have a few questions on my mind:
First: Why do we need to add alpha and gamma every time we sample the equation below? What if we delete the alpha and gamma from the equation? Would it still be possible to get the result?
Second: In LDA, we randomly assign a topic to every word in the document. Then, we try to optimize the topic by observing the data. Where is the part which is related to posterior inference in the equation above?
If you look at the inference derivation on Wiki, the alpha and beta are introduced simply because the theta and phi are both drawn from Dirichlet distribution uniquely determined by them separately. The reason of choosing Dirichlet distribution as the prior distribution (e.g. P(phi|beta)) are mainly for making the math feasible to tackle by utilizing the nice form of conjugate prior (here is Dirichlet and categorical distribution, categorical distribution is a special case of multinational distribution where n is set to one, i.e. only one trial). Also, the Dirichlet distribution can help us "inject" our belief that doc-topic and topic-word distribution are centered in a few topics and words for a document or topic (if we set low hyperparameters). If you remove alpha and beta, I am not sure how it will work.
The posterior inference is replaced with joint probability inference, at least in Gibbs sampling, you need joint probability while pick one dimension to "transform the state" as the Metropolis-Hasting paradigm does. The formula you put here is essentially derived from the joint probability P(w,z). I would like to refer you the book Monte Carlo Statistical Methods (by Robert) to fully understand why inference works.