Inference of Latent Class Models

In Infinite Latent Feature Models and the Indian Buffet Process, the authors point out that the Gibbs sampling inference in a latent class model depends on the following full conditional distribution of latent class c_i (Equation 17):

P(c_i = k | C_{-i}, X) \propto p(X|C) P(c_i=k | C_{-i})

The left-hand side of this equation is a general form. For LDA, it is P(z_i=k | Z_{-i}, W).

I am interested in the right-hand side, because it looks not trivial to compute p(X|C). Do we really need to compute it? I do not think so, because z_i is independent with W_{-i} given Z_{-i}.

An example of above statement is the derivation of LDA’s full conditional distribution, where we did the following expansion:
P(z_i = k | Z_{-i}, W) = \frac{P(z_i, Z_{-i}, w_i, W_{-i})}{P(Z_{-i}, w_i, W_{-i})} = \frac{P(z_i, Z_{-i}, w_i, W_{-i})}{P(Z_{-i}, W_{-i})}

In the right-most part of the equation, w_i is omitted because given Z_{-i}, it is only possible to generate W_{-i} but not w_i.

Continuing the derivation, we have

\frac{P(Z, W)}{P(Z_{-i}, W_{-i})} = P(z_i, w_i | Z_{-i}, W_{-i}) = p(w_i|z_i) p(z_i | Z_{-i})

This leads to the LDA Gibbs sampling rule appearing in the literature.  So Eqn. 17 can be rewritten as

P(c_i = k | C_{-i}, X) \propto p(x_i | c_i) P(c_i=k | C_{-i})