# Comparative Analysis of Exemplar-Based Approaches for Students' Learning Style Diagnosis Purposes
## 2 Methods
### Modelling Based on Bayesian Approach and/or Case-Based Reasoning
#### 2.5.4 Bayesian Case Model  
Bayesian case model is comprehensively presented by [PERSON] and her colleagues [8]. [PERSON] states that BCM is a general framework for Bayesian case-based reasoning (CBR) and prototype classification and clustering. It brings the intuitive power of CBR to a Bayesian generative framework, performing joint inference on clustering and explanations of the clusters. BCM captures dependencies among features via prototypes [8]. BCM is a discrete mixture model, i.e., it treats data as a mixture of several components. Each feature belongs to one of the components, and each component has a simple parametric form [8]. BCM is a generative statistical model that has clustering and explanation parts [8]. Thus, having N observations (X = {x_i,...,x_n}), each xi represents a random mixture over clusters. Generative models can specify a joint probability distribution over observed variables and latent variables, i.e., given an observable variable X and a target variable Y, it models the joint probability distribution on X \(\times\) Y. That is, a generative model is a model of the conditional probability of the observable X, given a target Y:p(X \(|\) Y = x). BCM generates each observation using the important pieces of related prototypes [8]. As in BCM, the underlying structure of the observations is represented by a standard discrete mixture model, each feature j of the observation i(x_i) comes from one of the clusters, the index of the cluster for xij is denoted by zij, and the full set of cluster assignments for observation-feature pairs is denoted by z. Each zij takes on the value of a cluster index between 1 and S [8]. BCM augments the standard mixture model with prototypes and subspace feature indicators that characterize the clusters, thus making BCM more interpretable. BCM allows sets of observations to be explained by latent variable z, i.e., to infer why some parts of the data are similar. In mixture models, the latent variable z corresponds to the mixture component, and it takes values in a discrete set; thus, p(z) is always a multinomial distribution. A graphical representation of BCM is presented in Figure 1[8]. The model is presented in Plate notation [44; 45]. Authors in [8] also presented the full BCM model in statistical form:  
\(\phi_{\text{sj}}\sim\text{Dirichlet}(\text{g}_{p_{\text{sj}},w_{\text{sj}}})\)\(\forall\) s, j;  
\(\text{g}_{p_{\text{sj}},w_{\text{sj}}}(\
u)=\lambda(1+\text{cl}_{\text{[}w_{ \text{sj}}=1\text{ and }p_{\text{sj}}=0]})\);  
\(\text{P}_{\text{s}}\sim\text{Uniform }(1,\text{ N})\forall\text{s}\);  
\(\text{wj}\sim\text{Bernoulli}(q)\forall\text{s}\), j;  
\(\pi_{\text{i}}\sim\text{Dirichlet}(\alpha)\forall\text{i}\);  
\(\text{z}_{\text{j}}\)\(\sim\)Multinomial (\(\pi_{\text{i}}\)) \(\forall\) i, j;  
\(\text{x}_{\text{j}}\)\(\sim\)Multinomial (\(\phi_{\text{zj}}\)) \(\forall\) i, j;  
As it is presented in the model, there are some hyperparameters (c, \(\lambda\), q) in BCM. Generally, parameters can help in defining or classifying a particular system. Parameters of the priori are called hyperparameters. It is a typical characteristic of conjugate priors that the dimensionality of the hyperparameters is one greater than that of the parameters of the original distribution. In the mixture model, using the Bayesian setting, the mixture weights and parameters themselves are random variables, and prior distributions will be placed over the variables. In BCM, the weights are typically viewed as a K-dimensional random vector drawn from a Dirichlet distribution (i.e., from the conjugate prior of the categorical distribution), and the parameters are distributed according to their respective conjugate priors. Authors in [8] emphasize that in BCM c and \(\lambda\) are constant hyperparameters that indicate how much the prototype will be copied in order to generate the observations. Setting c, \(\lambda\), and q can be done through cross-validation, another layer of hierarchy with more diffuse hyperparameters, or plain intuition.