Latent Dirichlet allocation - math words digest ? Latent Dirichlet allocation - is quite a popular topic in data-mining. 
Wikepedia mentions thousands citations in few years. 
Question 0 Can one give some digest for a math minded person of the key ideas ? 
Question 1 Is there non-trivial mathematical result behind  ?
Question 2 What practical problem does it help to solve ? 
Question 3 How the two questions above are related  ? 
Ideally I would be happy to see some theorem which would be important
for some practical problem. 
 A: Q1: De Finetti's theorem, that if the set of random variables $\{x_1,x_2,\ldots x_N \}$ is exchangeable, meaning their joint distribution $P(x_1,x_2,\ldots x_N)$ if invariant under permutation, then it can be represented as a mixture for some random variable $\theta$:
$$P(x_1,x_2,\ldots x_N)=\int d\theta\,p(\theta)\prod_{i=1}^N p(x_i|\theta).$$
The latent variable $\theta$ is the "L" from LDA. (The "D" refers to the, somewhat arbitrary, choice of a Dirichlet distribution for the $p$'s.)
De Finetti's theorem allows one to treat the joint distribution of words and topics in a document as conditionally independent and identically distributed with respect to an underlying "latent" parameter of a probability distribution. As the authors of LDA state in their 2003 paper, "Conditionally, the joint distribution of the random variables is simple and factored while marginally over the latent parameter, the joint distribution can be quite complex." 
The basic assumption that allows this efficient representation is the "bag-of-words" assumption, that the order of words in a document can be neglected when one tries to associate words with topics. 
I guess this also answers Q2 and Q3.
