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# The concept of topic models

The most known topic model, which is the one that I used throughout my thesis as well, is called Latent Dirichlet Allocation (LDA). A topic model is an algorithm that describes an iterative process, whose goal is to uncover the latent themes (“topics” in the topic modeling jargon) that are assumed to generate the words of a document collection. In this post I present some basic concepts of topic modeling.

The first question the above definition raises concerns these themes/topics: they are groups of words that tend to be semantically coherent and have different probabilities of appearance in text passages that discuss those topics. Imagine, for instance, a topic “Sports” and a topic “Chemistry”. Words like “ball”, “team” and “score” should have high probability of appearance when the topic is “Sports” and low probability when the topic is “Chemistry”. On the other hand, words like “mercury”, and “arsenic” should have high probability of appearance when the topic is “Chemistry”. The tricky part here is that words do not belong to only a single topic, they belong to every topic, but their probabilities are quite different. For some topics a particular word may appear with very high probability while for other with very low.

Therefore, there is an inherent assumption in topic models that there is a set of topic underlying a text collection (=group of documents). Moreover, each document is a mixture of these topics. This means that a document $d_1$ can be seen as

$d_i = 0.8\times\text{Sports} + 0.2\times\text{education}$

which would mean that 80% of the words of $d_1$ come from the Sports topic, while 20% from the Education topic. Once we have these mixture coefficients (0.8, 0.2 in the example), we can estimate a semantic similarity between documents or enable other types of applications.

Of course, there is a catch in this. The problem is that we only observe the documents. Therefore, we need an algorithm for identifying (i) the topics (semantically coherent group of words) and, (ii) which topics occur in a document and in what extent. The (i) point means we need to identify in some way that words like[ball, team] and [mercury, and arsenic] are semantically coherent. To do this we make use of the fact that words to tend to co-occur in similar contexts, should be similar (paraphrasing the expression “show me your friend and I will tell you who you are”). Once we achieve (i), (ii) can be seen as a by-product of it.

This process is also shown in Table 1. The Table illustrates the words that constitute five topics, when LDA is applied on a set of Wikipedia documents. Note, here, that the titles Science, Art, Cinema, etc. were selected manually. The algorithm only returns the words that constitute the tweets.

Science Art Cinema Music Elections
university art film record election
research new role music canadian
science york televis band serve
professor paint role album party

# Inference

As discussed above, the inference problem of topic models is to discover the topics. In reality, we only have access to the words that constitute the documents and we do not directly observe topics. The topic modeling objective is to extrapolate from the statistical behavior and co-occurrence of words and construct the coherent groups of words. Unfortunately, there is no way to infer the topics exactly: there are too many unknowns in the beginning. To overcome this problem we rely on Markov Chain Monte Carlo sampling mechanisms. To avoid confusion I will simply describe the rationale behind them.

To understand how inference works let us pretend for a moment that we had the problem mostly solved. Consider the toy example where we know the topics of every words and document and there is a single word, the last one, that remains and we need to find its topic. It is the word player for example. The question is “Which is the topic of player” ? Or even better: What is the probability that player was generated by the topic $k$? Although we can not know for sure, we can answer the questions while considering two facts: (a) has player been generated by the topic $k$ previously in out collection? and (b) Is topic $k$ a common topic in the rest of the document?

The idea behind question (a) is that if player often occurs in discussions of $k$, then this instance of player might as well have high probability to belong to $k$. But a word can be common in more than one topic. And we don’t want to assign player to a topic about sports if this document is mostly about Elections and to overcome this we also need to consider the answer of question (b).

To formulate (a) and (b), we consider the probability that stems from the frequency of this word type (player) with topic $k$ by the number of other words in document $d$ that already belong to $k$. The result will represent the probability that this word came from $k$. The actual formula (the only formula in this document):

$p(\text{word w has topic } k | W,D)= \frac{\text{\# of times w appears with k}+\beta}{\text{Total tokens in } k+\beta}$

where $\beta$ is a hyper-parameter, that is a variable whose value we need to tune and $W,D$ refer to the rest of the words and topic assigned to them in the corpus. Repeating this process in the scale of the document collection allows us to uncover both the topic and the topic distributions of the documents. Of course, there are long mathematical proofs of the convergence of this process that I do not detail here.

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