Latent Dirichlet allocation

In natural language processing, latent Dirichlet allocation (LDA) is a Bayesian network (and, therefore, a generative statistical model) for modeling automatically extracted topics in textual corpora. The LDA is an example of a Bayesian topic model. In this, observations (e.g., words) are collected into documents, and each word's presence is attributable to one of the document's topics. Each document will contain a small number of topics.

History

In the context of population genetics, LDA was proposed by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. LDA was applied in machine learning by David Blei, Andrew Ng and Michael I. Jordan in 2003.

Overview

Evolutionary biology and bio-medicine

In evolutionary biology and bio-medicine, the model is used to detect the presence of structured genetic variation in a group of individuals. The model assumes that alleles carried by individuals under study have origin in various extant or past populations. The model and various inference algorithms allow scientists to estimate the allele frequencies in those source populations and the origin of alleles carried by individuals under study. The source populations can be interpreted ex-post in terms of various evolutionary scenarios. In association studies, detecting the presence of genetic structure is considered a necessary preliminary step to avoid confounding.

Clinical psychology, mental health, and social science

In clinical psychology research, LDA has been used to identify common themes of self-images experienced by young people in social situations. Other social scientists have used LDA to examine large sets of topical data from discussions on social media (e.g., tweets about prescription drugs). Additionally, supervised Latent Dirichlet Allocation with covariates (SLDAX) has been specifically developed to combine latent topics identified in texts with other manifest variables. This approach allows for the integration of text data as predictors in statistical regression analyses, improving the accuracy of mental health predictions. One of the main advantages of SLDAX over traditional two-stage approaches is its ability to avoid biased estimates and incorrect standard errors, allowing for a more accurate analysis of psychological texts. In the field of social sciences, LDA has proven to be useful for analyzing large datasets, such as social media discussions. For instance, researchers have used LDA to investigate tweets discussing socially relevant topics, like the use of prescription drugs. By analyzing these large text corpora, it is possible to uncover patterns and themes that might otherwise go unnoticed, offering valuable insights into public discourse and perception in real time.

Musicology

In the context of computational musicology, LDA has been used to discover tonal structures in different corpora.

Machine learning

One application of LDA in machine learning - specifically, topic discovery, a subproblem in natural language processing – is to discover topics in a collection of documents, and then automatically classify any individual document within the collection in terms of how "relevant" it is to each of the discovered topics. A topic is considered to be a set of terms (i.e., individual words or phrases) that, taken together, suggest a shared theme. For example, in a document collection related to pet animals, the terms dog, spaniel, beagle, golden retriever, puppy, bark, and woof would suggest a DOG_related theme, while the terms cat, siamese, Maine coon, tabby, manx, meow, purr, and kitten would suggest a CAT_related theme. There may be many more topics in the collection – e.g., related to diet, grooming, healthcare, behavior, etc. that we do not discuss for simplicity's sake. (Very common, so called stop words in a language – e.g., "the", "an", "that", "are", "is", etc., – would not discriminate between topics and are usually filtered out by pre-processing before LDA is performed. Pre-processing also converts terms to their "root" lexical forms – e.g., "barks", "barking", and "barked" would be converted to "bark".) If the document collection is sufficiently large, LDA will discover such sets of terms (i.e., topics) based upon the co-occurrence of individual terms, though the task of assigning a meaningful label to an individual topic (i.e., that all the terms are DOG_related) is up to the user, and often requires specialized knowledge (e.g., for collection of technical documents). The LDA approach assumes that: When LDA machine learning is employed, both sets of probabilities are computed during the training phase, using Bayesian methods and an Expectation Maximization algorithm. LDA is a generalization of older approach of probabilistic latent semantic analysis (pLSA), The pLSA model is equivalent to LDA under a uniform Dirichlet prior distribution. pLSA relies on only the first two assumptions above and does not care about the remainder. While both methods are similar in principle and require the user to specify the number of topics to be discovered before the start of training (as with K-means clustering) LDA has the following advantages over pLSA:

Model

With plate notation, which is often used to represent probabilistic graphical models (PGMs), the dependencies among the many variables can be captured concisely. The boxes are "plates" representing replicates, which are repeated entities. The outer plate represents documents, while the inner plate represents the repeated word positions in a given document; each position is associated with a choice of topic and word. The variable names are defined as follows: The fact that W is grayed out means that words w_{ij} are the only observable variables, and the other variables are latent variables. As proposed in the original paper, a sparse Dirichlet prior can be used to model the topic-word distribution, following the intuition that the probability distribution over words in a topic is skewed, so that only a small set of words have high probability. The resulting model is the most widely applied variant of LDA today. The plate notation for this model is shown on the right, where K denotes the number of topics and are V-dimensional vectors storing the parameters of the Dirichlet-distributed topic-word distributions (V is the number of words in the vocabulary). It is helpful to think of the entities represented by \theta and \varphi as matrices created by decomposing the original document-word matrix that represents the corpus of documents being modeled. In this view, \theta consists of rows defined by documents and columns defined by topics, while \varphi consists of rows defined by topics and columns defined by words. Thus, refers to a set of rows, or vectors, each of which is a distribution over words, and refers to a set of rows, each of which is a distribution over topics.

Generative process

To actually infer the topics in a corpus, we imagine a generative process whereby the documents are created, so that we may infer, or reverse engineer, it. We imagine the generative process as follows. Documents are represented as random mixtures over latent topics, where each topic is characterized by a distribution over all the words. LDA assumes the following generative process for a corpus D consisting of M documents each of length N_i:

Choose, where and is a Dirichlet distribution with a symmetric parameter \alpha which typically is sparse (\alpha < 1)
Choose, where and \beta typically is sparse
For each of the word positions i, j, where, and (Note that multinomial distribution here refers to the multinomial with only one trial, which is also known as the categorical distribution.) The lengths N_i are treated as independent of all the other data generating variables (w and z). The subscript is often dropped, as in the plate diagrams shown here.

Definition

A formal description of LDA is as follows: We can then mathematically describe the random variables as follows:

Inference

Learning the various distributions (the set of topics, their associated word probabilities, the topic of each word, and the particular topic mixture of each document) is a problem of statistical inference.

Monte Carlo simulation

The original paper by Pritchard et al. used approximation of the posterior distribution by Monte Carlo simulation. Alternative proposal of inference techniques include Gibbs sampling.

Variational Bayes

The original ML paper used a variational Bayes approximation of the posterior distribution.

Likelihood maximization

A direct optimization of the likelihood with a block relaxation algorithm proves to be a fast alternative to MCMC.

Unknown number of populations/topics

In practice, the optimal number of populations or topics is not known beforehand. It can be estimated by approximation of the posterior distribution with reversible-jump Markov chain Monte Carlo.

Alternative approaches

Alternative approaches include expectation propagation. Recent research has been focused on speeding up the inference of latent Dirichlet allocation to support the capture of a massive number of topics in a large number of documents. The update equation of the collapsed Gibbs sampler mentioned in the earlier section has a natural sparsity within it that can be taken advantage of. Intuitively, since each document only contains a subset of topics K_d, and a word also only appears in a subset of topics K_w, the above update equation could be rewritten to take advantage of this sparsity. In this equation, we have three terms, out of which two are sparse, and the other is small. We call these terms a, b and c respectively. Now, if we normalize each term by summing over all the topics, we get: Here, we can see that B is a summation of the topics that appear in document d, and C is also a sparse summation of the topics that a word w is assigned to across the whole corpus. A on the other hand, is dense but because of the small values of \alpha & \beta, the value is very small compared to the two other terms. Now, while sampling a topic, if we sample a random variable uniformly from, we can check which bucket our sample lands in. Since A is small, we are very unlikely to fall into this bucket; however, if we do fall into this bucket, sampling a topic takes O(K) time (same as the original Collapsed Gibbs Sampler). However, if we fall into the other two buckets, we only need to check a subset of topics if we keep a record of the sparse topics. A topic can be sampled from the B bucket in O(K_d) time, and a topic can be sampled from the C bucket in O(K_w) time where K_d and K_w denotes the number of topics assigned to the current document and current word type respectively. Notice that after sampling each topic, updating these buckets is all basic O(1) arithmetic operations.

Aspects of computational details

Following is the derivation of the equations for collapsed Gibbs sampling, which means \varphis and \thetas will be integrated out. For simplicity, in this derivation the documents are all assumed to have the same length N_{}. The derivation is equally valid if the document lengths vary. According to the model, the total probability of the model is: where the bold-font variables denote the vector version of the variables. First, and need to be integrated out. All the \thetas are independent to each other and the same to all the \varphis. So we can treat each \theta and each \varphi separately. We now focus only on the \theta part. We can further focus on only one \theta as the following: Actually, it is the hidden part of the model for the j^{th} document. Now we replace the probabilities in the above equation by the true distribution expression to write out the explicit equation. Let n_{j,r}^i be the number of word tokens in the j^{th} document with the same word symbol (the r^{th} word in the vocabulary) assigned to the i^{th} topic. So, n_{j,r}^i is three dimensional. If any of the three dimensions is not limited to a specific value, we use a parenthesized point (\cdot) to denote. For example, denotes the number of word tokens in the j^{th} document assigned to the i^{th} topic. Thus, the right most part of the above equation can be rewritten as: So the \theta_j integration formula can be changed to: The equation inside the integration has the same form as the Dirichlet distribution. According to the Dirichlet distribution, Thus, Now we turn our attention to the part. Actually, the derivation of the part is very similar to the part. Here we only list the steps of the derivation: For clarity, here we write down the final equation with both and integrated out: The goal of Gibbs Sampling here is to approximate the distribution of. Since is invariable for any of Z, Gibbs Sampling equations can be derived from directly. The key point is to derive the following conditional probability: where Z_{(m,n)} denotes the Z hidden variable of the n^{th} word token in the m^{th} document. And further we assume that the word symbol of it is the v^{th} word in the vocabulary. denotes all the Zs but Z_{(m,n)}. Note that Gibbs Sampling needs only to sample a value for Z_{(m,n)}, according to the above probability, we do not need the exact value of but the ratios among the probabilities that Z_{(m,n)} can take value. So, the above equation can be simplified as: Finally, let be the same meaning as n_{j,r}^i but with the Z_{(m,n)} excluded. The above equation can be further simplified leveraging the property of gamma function. We first split the summation and then merge it back to obtain a k-independent summation, which could be dropped: Note that the same formula is derived in the article on the Dirichlet-multinomial distribution, as part of a more general discussion of integrating Dirichlet distribution priors out of a Bayesian network.

Contents