In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can be described by a set of unobserved "topics." For example, given a set of news articles, LDA might discover that one topic is characterized by words like "president", "government", and "election", while another is characterized by "team", "game", and "score". It is one of the most common topic models. The LDA model was first presented as a graphical model for population genetics by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. The model was subsequently applied to machine learning by David Blei, Andrew Ng, and Michael I. Jordan in 2003. Although its most frequent application is in modeling text corpora, it has also been used for other problems, such as in clinical psychology, social science, and computational musicology. The core assumption of LDA is that documents are represented as a random mixture of latent topics, and each topic is characterized by a probability distribution over words. The model is a generalization of probabilistic latent semantic analysis (pLSA), differing primarily in that LDA treats the topic mixture as a Dirichlet prior, leading to more reasonable mixtures and less susceptibility to overfitting. Learning the latent topics and their associated probabilities from a corpus is typically done using Bayesian inference, often with methods like Gibbs sampling or variational Bayes. == 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 == === Population genetics === In population genetics, 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 and cultural differences in China. 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: The semantic content of a document is composed by combining one or more terms from one or more topics. Certain terms are ambiguous, belonging to more than one topic, with different probability. (For example, the term training can apply to both dogs and cats, but are more likely to refer to dogs, which are used as work animals or participate in obedience or skill competitions.) However, in a document, the accompanying presence of specific neighboring terms (which belong to only one topic) will disambiguate their usage. Most documents will contain only a relatively small number of topics. In the collection, e.g., individual topics will occur with differing frequencies. That is, they have a probability distribution, so that a given document is more likely to contain some topics than others. Within a topic, certain terms will be used much more frequently than others. In other words, the terms within a topic will also have their own probability distribution. 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: LDA yields better disambiguation of words and a more precise assignment of documents to topics. Computing probabilities allows a "generative" process by which a collection of new "synthetic documents" can be generated that would closely reflect the statistical characteristics of the original collection. Unlike LDA, pLSA is vulnerable to overfitting especially when the size of corpus increases. The LDA algorithm is more readily amenable to scaling up for large data sets using the MapReduce approach on a computing cluster. == 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: M denotes the number of documents N is number of words in a given document (document i has N i {\displaystyle N_{i}} words) α is the parameter of the Dirichlet prior on the per-document topic distributions β is the parameter of the Dirichlet prior on the per-topic word distribution θ i {\displaystyle \theta _{i}} is the topic distribution for document i φ k {\displaystyle \varphi _{k}} is the word distribution for topic k z i j {\displaystyle z_{ij}} is the topic for the j-th word in document i w i j {\displaystyle w_{ij}} is the specific word. The fact that W is grayed out means that words w i j {\displaystyle 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 to
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