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When studying the latent Dirichlet allocation, I am not very clear about some procedures in their deriving equations. Please refer to the attached figure, how to understand those two steps, marked as 1 and 2 in the figure. enter image description here

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1: denoting $z_{d,n} = A$, $w_{d,n} = B$ , $\theta_{d} = C$, $\beta_k = D$, and using the Bayes rule:

$\displaystyle p(A|B,C,D) = \frac{p(A|C,D) \ p(B|A,C,D)}{p(B|C,D)} \propto p(A|C,D)\ p(B|A,C,D)$,

where $\propto$ means it is proportional, which is the case because $p(B|C,D)$ does not depend on $A$ i.e. is a constant.

2: we can get knowing that both $p(z|\theta)$ and $p(w|z,\beta)$ are (presumably) discrete distributions, thus assigning fixed values $\theta_{d,k}$ and $\beta_{k, w_{d,n}}$.

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  • $\begingroup$ I once heard that proportional will not affect sampling at all, and that's why it is frequently used in Bayesian statistics. How to understand that? Thanks. $\endgroup$ – user3269 Mar 14 '13 at 18:31
  • $\begingroup$ I would say that if you are working with probability distributions, constant factors can be neglected since you can renormalize the distribution to sum up to 1. $\endgroup$ – psycharo Mar 14 '13 at 19:08

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