# Comparative Analysis of Exemplar-Based Approaches for Students' Learning Style Diagnosis Purposes
## 2 Methods
### Modelling Based on Bayesian Approach and/or Case-Based Reasoning
#### 2.5.2 Latent Dirichlet Allocation-Mixed Membership Model  
As the research includes Bayesian methods that may be used for students' learning style modelling, first, we briefly describe Latent Dirichlet allocation, which is a generative model for collections of discrete data, a sophisticated application of the Bayesian technique. As typically LDA is used in topic models to classify text in a document to a particular topic, LDA is described using a topic modelling example. Then, it is adapted to learning style modelling by analogy. Sections 2.5.3 and 2.6.1 explain how the LDA approach may be used to model learning style. In topic modelling, documents are represented as random mixtures over latent topics, where each topic is characterized by a distribution over all the words. LDA builds a topic per document model and words per topic model, modeled as Dirichlet distributions. Each document is modeled as a multinomial distribution of topics, and each topic is modeled as a multinomial distribution of words. LDA assumes that documents are produced from a mixture of topics. Preprocessing of the raw text, conversion of text to a bag of words, and specification of how many topics are in the data set are made before LDA is performed [38]. As [15] explains, LDA works by first making a key assumption: the way a document was generated was by picking a set of topics and then for each topic picking a set of words. So, LDA reverse-engineers this process. LDA assumes that the prior is arising from theDirichlet distribution. At each iteration, the posterior is updated in order to reflect more proper topics. To find groups of tightly co-occurring words, LDA has a trade-off two goals: for each document, allocate its words to as few topics as possible, and for each topic, assign high probability to as few terms as possible [39]. In LDA, it is a must to specify the number of topics first. Thus, we have the probability distribution of topics in documents, which indicates how much the document \"likes\" the topic, and the probability distribution of words in topics indicates how much a topic \"likes\" a word. Following [40], in LDA, we assume that words are generated by topics (by fixed conditional distributions) and that those topics are infinitely exchangeable within the document. An infinite sequence of random variables is infinitely exchangeable if every finite subsequence is exchangeable. [PERSON]'s representation theorem states that exchangeable observations are conditionally independent relative to some latent variable, and an epistemic probability distribution could then be assigned to this variable. Authors of [40] refine that the joint distribution of an infinitely exchangeable sequence of random variables is as if a random parameter was drawn from some distribution; then the random variables in question were independent and identically distributed, conditioned on that parameter. By [PERSON]'s theorem, the probability of a sequence of words and topics must therefore have the form:  
\[\mathrm{p}(\mathrm{w},\mathrm{z})=\int\mathrm{p}(\theta)\prod\mathrm{p}( \mathrm{z}_{\mathrm{n}}|\theta)\mathrm{p}(\mathrm{w}_{\mathrm{n}}|\mathrm{z} _{\mathrm{n}})\mathrm{d}\theta \tag{5}\]  
\(\theta\) is the random parameter of a multinomial distribution over topics, and \(\mathrm{w}\) is a document in the document corpus. The text of a document is treated as bag of words disregarding grammar and even word order but keeping multiplicity. \(\mathrm{z}\) is a hidden topic variable. LDA distribution on documents is obtained by marginalizing out the topic variables and endowing \(\theta\) with a Dirichlet distribution [40]. Using LDA for making inference, it is needed to compute the posterior distribution of the hidden variables given a document:  
\[\mathrm{p}(\theta,\mathrm{z}|\mathrm{w},\alpha,\beta)=\frac{\mathrm{p}(\theta,\mathrm{z},\mathrm{w}|\alpha,\beta)}{\mathrm{p}(\mathrm{w}|\alpha,\beta)} \tag{6}\]  
Parameters \(\alpha\) and \(\beta\) are corpus-level parameters, assumed to be sampled once in the process of generating a corpus.