###LDA does not have a distance metric

The intuition behind the LDA topic model is that words belonging to a topic appear together in documents. Unlike typical clustering algorithms like K-Means, it does not assume any distance measure between topics. Instead it infers topics purely based on word counts. 

This can be appreciated from the Gibbs sampler described in paper by [Griffiths et al.](http://psiexp.ss.uci.edu/research/papers/sciencetopics.pdf):

$$
P(z_i=j \mid \textbf{z}_{-i} , \textbf{w} ) \propto \frac{n^{(w_i)}_{-i,j}+\beta}{n^{(.)}_{-i,j}+W\beta} \times \frac{n^{(d_i)}_{-i,j}+\alpha}{n^{(d_i)}_{-i,.}+T\alpha}
$$

$P(z_i=j \mid \textbf{z}_{-i} , \textbf{w} )$ refers to the probability of assigning topic $j$ to $i^{th}$ word, given all other assignments. This depends on two probabilities:

 1. Probability of word $w_i$ in topic $j$
 2. Probability of topic $j$ in document $d_i$

These probabilities can be easily computed using the following counts: 

 - $n^{(w_i)}_{-i,j}:$ number of times word $w_i$ was assigned to topic $j$
 - $n^{(.)}_{-i,j}:$ total number of words assigned to topic $j$
 - $n^{(d_i)}_{-i,j}:$ number of times topic $j$ was assigned in document $d_i$
 - $n^{(d_i)}_{-i,.}:$ total number of topics assigned in document $d_i$
 - $T:$ number of topics
 - $W:$ number of words in vocabulary
 - $\alpha, \beta:$ Dirichlet hyperparameters

Note that all counts are excluding the current assignment, denoted by the $-i$ subscript.

----
###Why does LDA work? 

Referring to these [Video Lectures](http://videolectures.net/mlss09uk_blei_tm/), David Blei attributes it to the following:

[![Why LDA works][1]][1]


  [1]: https://i.sstatic.net/Sf4y5.png