sub-simplex in LDA

Mathematically speaking a simplex is the generalization of a triangle.

It has the really nice property that the parameter of the vertex has to sum to one, like a probability ditribution, which is desirable. A Dirichlet Distribution is said to be a distribution over the simplex, in terms that it will give different weights to the corners of the triangle, and this weights have to sum to one.

The word simplex is the dictionary of every possible word, and the distribution over this dictionary has to sum to 1, now, the topic-simplex is also defined over all the words, but is not every possible distribution like in the previous case, but very specific distribution over words, so they are a subset of the simplex of words.

An example:

Suppose you have your dictionary: {Dog, Science, Math, Cat}

Your simplex of words is the space of probability distribution that can define the discrete weights of this.

• Dog=0.9, Science=0.05, Math=0.05, Cat=0 >> This is one member of the simplex.

Now the topics are going to be the same kind of distributions, but the weight will make more sense:

• Dog=0.4, Science=0.1, Math=0.1, Cat=0.4 > Perhaps a topic on scientific subjects

• Dog=0.1, Science=0.4, Math=0.4, Cat=0.1 > Perhaps a topic on animals

Both of the past examples are parts of the overall space, but they define topics, which the first one of the examples did not.