## 2 Materials and Methods
### Proposed Model
\[\hat{y}=P(y|X)=\frac{1}{P(y)^{n-1}}\prod_{i=0}^{n}P(y|X_{i})\left(\frac{P(X_{i+1}| \bigcap_{i=0}^{i}X_{\mu})}{P(X_{i+1})}\right)^{-1}\]5  
Using Equation (18), the class likelihoods will be  
\[\hat{y}=P(y|X)=\frac{1}{P(y)^{n-1}}\prod_{i=0}^{n}P(y|X_{i})\left(\frac{P(X_{i+1}| \bigcap_{i=0}^{i}X_{\mu})}{P(X_{i+1})}\right)^{-1}\]6  
\[\hat{y}=P(y|X)=\frac{1}{P(y)^{n-1}}\prod_{i=0}^{n}P(y|X_{i})\left(\frac{P(X_{i+1}| \bigcap_{i=0}^{i}X_{\mu})}{P(X_{i+1})}\right)^{-1}\]7  
\[\hat{y}=P(y|X)=\frac{1}{P(y)^{n-1}}\prod_{i=0}^{n}P(y|X_{i})\left(\frac{P(X_{i+1}| \bigcap_{i=0}^{i}X_{\mu})}{P(X_{i+1})}\right)^{-1}\]8  
\[\hat{y}=P(y|X)=\frac{1}{P(y)^{n-1}}\prod_{i=0}^{n}P(y|X_{i})\left(\frac{P(X_{i+1}| \bigcap_{i=0}^{i}X_{\mu})}{P(X_{i+1})}\right)^{-1}\]9  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]0  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]1  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]2  
Using Equation (19) and ordering the features to be conditionally dependent from \(x_{1}\) to \(x_{7}\) with a Markov property of a level 1 assumption, the observation likelihoods will be  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]3  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]4  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]5  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]6  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]7  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]8  
This implies that, using the Bernoulli version of our proposed model, the classification of a corpus given that it has the words economy, crises and leaders, will be,  
\[\hat{y}_{k}=\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}P(y_{k }|X)\]9  
\[=H(X)\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}\left(\frac{1 }{P(y_{k})^{n-1}}\prod_{i=0}^{n}P(y_{k}|X_{i})\right) \tag{13}\]0  
Similar to the classification results using conventional models, our proposed model also classifies the statement as Business news but with larger floating point value than the conventional model. Furthermore, the heterogeneity between the features at the last prediction instance can be estimated using the heterogeneity function \(H(X)\) as follows:  
\[=H(X)\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}\left(\frac{1 }{P(y_{k})^{n-1}}\prod_{i=0}^{n}P(y_{k}|X_{i})\right) \tag{13}\]1  
We consider the reciprocal of this value as a form of mutuality between the features, which measures their similarity (i.e., homogeneity), and can be used to indirectly measure their heterogeneity in this model. Thus, the mutuality \(M(X)\) between the features will be,  
\[=H(X)\operatorname*{arg\,max}_{k\in\{1,2,\ldots,m\}}\left(\frac{1 }{P(y_{k})^{n-1}}\prod_{i=0}^{n}P(y_{k}|X_{i})\right) \tag{13}\]2  
Therefore, if \(H(X)=0.001513\), then \(M(X)=660.983143\). This implies that there is more probabilistic similarity than dissimilarity between the features. In other words, the features are probabilistically more joined together than disjoint in their occurrence. In this study, \(M(X)\) measures the correlation (or association) between the features in terms of the probabilistic similarity (i.e., homogeneity) of their dependency on one another. For any two features \(X_{i}\) and \(X_{j}\), where each one is conditioned on the other, if \(M(X_{i},X_{j})=1\), then \(P(X_{i}|X_{j})=P(X_{i})\) and \(P(X_{j}|X_{i})=P(X_{j})\), which implies \(X_{i}\) and \(X_{j}\) are probabilistically identical, but non-similar in their dependence to each other. If \(M(X)>1\), then \(P(X_{i}|X_{j})>P(X_{i})\) and \(P(X_{j}|X_{i})>P(X_{j})\), which implies \(X_{i}\) and \(X_{j}\) have a direct (i.e., increase in value when conditioned) probabilistic similarity in their dependency to one another.