## 2 Materials and Methods
### Proposed Model  
Unlike the conventional approach that makes use of all the features of the text corpus to infer and learn the given category, our proposed approach provides the possibility to segment the input features into multiple groups of input features, forming different corpora, on which separate learning and inference are performed. This model is defined mathematically using probabilistic logic as follows:  
**Axiom 1. **  
\[P(X_{i}|X,y)=P(X_{i}|y) \tag{11}\]  
**Proposition 1**. : \[\hat{y}=P(y|X)=\frac{1}{P(y)^{n-1}}\prod_{i=0}^{n}P(y|X_{i})\left(\frac{P(X_{i+1}| \bigcap_{i=0}^{i}X_{\mu})}{P(X_{i+1})}\right)^{-1}\] (12)  
where \(y\) is the given set of categories, \(X\) is a vector of feature vectors \(X=(X_{0},X_{1},\ldots,X_{n})\) and \(X_{i}=(x_{1},x_{2},\ldots)\), \(\hat{y}\) is the posterior distribution (i.e., class posterior) of \(y\) given \(X\), \(P(y)\) is the prior distribution (i.e., class prior) of \(y\) based on \(X\), \(P(y|X_{i})\) is the partial posterior distribution (i.e., partial class posterior) of \(y\) given \(X_{i}\), \(P(X_{i+1})\) is the prior distribution (i.e., observation prior) of \(X_{i+1}\) based on \(\bigcap_{i=0}^{i}X_{\mu}\), \(P(X_{i}|\bigcap_{i=0}^{i}X_{\mu})\) is the posterior distribution (i.e., observation posterior) of \(X_{i}\) given \(\bigcap_{i=0}^{i}X_{\mu}\), and \(n\) is the number of features vectors. It is worth noting that the posterior probability distributions can be interpreted as likelihood functions, i.e., the posterior of \(y\) given \(X\) is similar to the likelihood of \(X\) given \(y\), and so on. Thus, the term posterior is used interchangeably with the term likelihood in this manuscript. The proof of Proposition 1 is given in Appendix A. The inference and learning based on this proposed model can then be expressed as  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\] \[=H(X)\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}\left(\frac{1 }{P(y_{k})^{n-1}}\prod_{i=0}^{n}P(y_{k}|X_{i})\right) \tag{13}\] \[\Rightarrow\hat{y}_{k}\propto\operatorname*{arg\,max}_{k\in\{1, 2,\ldots,m\}}\left(\frac{1}{P(y_{k})^{n-1}}\prod_{i=0}^{n}P(y_{k}|X_{i})\right)\] (14) \[\max(\hat{y})=\operatorname*{arg\,max}_{j\in\{1,2,\ldots,J\}}P(y _{k}^{(j)}|X^{(j)}) \tag{15}\]  
where \(H(X)=\prod_{i=0}^{n}\left(\frac{P(X_{i+1}|\bigcap_{y=0}^{i}X_{\mu})}{P(X_{i+1 })}\right)^{-1}\), \(H(X)\in\,[0,\infty]\)  
During inference and learning, estimating the priors and likelihoods (i.e., posteriors) based on this model for both the class and observation is given as  
Class prior: \[P(c)=\frac{n_{c}}{n}\] (16) Observation prior: \[P(w)=\frac{n_{w}}{n}\] (17) Class Likelihood: \[P(c|w)=\frac{n_{c,w}+\alpha|V|}{n_{w}+\alpha|V|}\,\ \ \alpha=1\] (18) Observation Likelihood: \[P(w_{i}|w_{j})=\frac{n_{w_{i},w_{j}}+\alpha}{n_{w_{j}}+\alpha|V|}\,\ \ \alpha=1\] (19)  
where \(n\) is the total number of instances in the text vectorization, \(c\) is a class, \(n_{c}\) is the frequency (number of) occurrences of a class, \(w\) is a feature vector, \(n_{w}\) is the frequency of occurrences of a feature vector \(w\), \(n_{c,w}\) is the frequency of occurrence of a class \(c\) given the occurrence of a feature vector \(w\), \(n_{w_{i},w_{j}}\) is the frequency of occurrence of a feature vector \(w_{j}\) given the occurrence of another feature vector \(w_{j}\), \(\alpha\) is a smoothing parameter, which is equal to 1 for Laplacian smoothing, and \(V\) is the number of vectorized features. Using this proposed model, for any given classification task, the first step is to segment the features into parts, forming separate feature vectors, then inference and learning are applied on each of the feature vectors using the class likelihood (or partial class posterior) \(P(c|w)\). The partial class posterior can also be expressed as an inference problem on each segmented feature where our model or a Bayesian model can be applied to generate its results.