# Movie Recommendation through Multiple Bias Analysis +
## 3 MBA: Multiple Bias Analysis
### Vanilla Model  
From Figure 1, MBA was expanded to reflect the generalized feature (GF) and personalized feature (PF) on BBP, and Equation (22) shows the rating prediction of the Vanilla model. \[P_{wt}^{VM}(s,u,i)=s+b_{wt}(s,u)+b_{wt}(s,i)+GF(s,u,i)+PF(s,u,i) \tag{22}\]  
The \(wt\) and \(s\) that are used in Equation (22) are the weight type and pinpoint score introduced in Section 2.2. MBA optimizes \(s\) using Algorithm 1 in the preprocessing stage  
Figure 1: Summary of Multiple Bias Analysis (MBA). before computing GF and PF. GF means a bias (global bias) sympathized with other users, and PF means a bias (local bias) independent of other users. Regarding [PERSON], GF refers to the public's tendency toward [PERSON], and PF refers to the personal tendency toward [PERSON]. GF represents the tendency of a feature that is sympathized with other users. The GF-Score was computed, as shown in Equation (23), and it was designed to reflect the distance \(\omega\) between the feature bias \(b(s,f)\) and user \(u\) computed from all users. \[ \begin{split} GF(s,u,i)&=\frac{1}{|\textit{UF}(u)|} \sum_{f\in\textit{UF}(u)}\omega(u,f)\cdot b(s,f)\\ &\quad+\frac{1}{|\textit{IF}(i)|}\sum_{T\in\textit{TF}(i)}\frac{1} {|\textit{TF}|}\sum_{f\in\textit{TF}}\omega(i,f)\cdot b(s,f)\\ &=\frac{1}{|\textit{UF}(u)|}\sum_{f\in\textit{UF}(u)}\omega(u,f) \cdot\left(\frac{\sum_{o,j\in R(f)}r(o,j)}{|R(f)|}-s\right)\\ &\quad+\frac{1}{|\textit{IF}(i)|}\sum_{TF\subset\textit{IF}(i)} \frac{1}{|\textit{TF}|}\sum_{f\in\textit{TF}}\omega(i,f)\cdot\left(\frac{\sum_{ o,j\in R(f)}r(v,j)}{|R(f)|}-s\right)\end{split}  \tag{23}\]  
\(\textit{UF}(u)\) is a feature set for the user \(u\)'s age, gender, and occupation, and \(\textit{IF}(i)\) is a feature set for the actor, director, and genre of item \(i\). Here, because the elements of \(\textit{IF}(i)\) are a set (actor, director, genre), it was designed to add the average of the target feature \(TF\). \(R(f)\) is the set of rated scores belonging to feature \(f\), \(r(v,j)\) is the rated score for item \(j\) of user \(v\), \(\omega(u,f)\) is the weight of feature \(f\) of user \(u\), and \(\omega(i,f)\) is the weight of feature \(f\) of item \(i\). The range of the values of the feature weight \(\omega\) is \(-1\) to \(1\), and the value that is computed through Equation (26) is used in Equations (28) and (29). In GF, \(\omega\) is optimized to compute how close the feature \(f\) was to the popular bias. If user \(u\) is a _30-year-old male student_, _UF(u)=[Male, 30(Group-3), Student]_ will be used. The first term of GF can be disjoined and written, as follows:  
\[ \begin{split}\frac{1}{|\textit{UF}(u)|}\sum_{f\in\textit{UF}(u)} \omega(u,f)\cdot b(s,f)=&\omega(u,AgeGroup(u))\cdot b(s,AgeGroup (u))\\ +&\omega(u,Gender(u))\cdot b(s,Gender(u))\\ +&\omega(u,Occupation(u))\cdot b(s,Occupation(u))) \end{split} \]  
If item \(i\) is the movie _The Truman Show_, _IF(i)=[([PERSON], \(\cdots\)), [PERSON]_, (Comedy, \(\cdots\) )]_ will be used. The second term of GF can be disjoined and written, as follows:  
\[ \begin{split}\frac{1}{|\textit{IF}(i)|}\sum_{TF\subset\textit{ IF}(i)}\frac{1}{|\textit{TF}|}\sum_{f\in\textit{TF}}\omega(i,f)\cdot b(s,f)& =\frac{1}{|\textit{Actors}(i)|}\sum_{f\in\textit{Actors}(i)} \omega(i,f)\cdot b(s,f)\\ +&\frac{1}{|\textit{Director}(i)|}\sum_{f\in\textit{ Director}(i)}\omega(i,f)\cdot b(s,f)\\ +&\frac{1}{|\textit{Genres}(i)|}\sum_{f\in\textit{ Genres}(i)}\omega(i,f)\cdot b(s,f)\end{split} \]  
When feature \(f\) is [PERSON], the \(b(s,f)\) that is used in GF-Score is the bias (global bias) of [PERSON] regarding all users, and \(\omega(u,f)\) is the distance between user \(u\) and \(b(s,f)\) (or importance). PF represents a personalized tendency. The PF-Score was designed to be computed, as shown in Equation (24), and it reflected the reliability \(\omega\) for feature bias \(b(s,f)\) and \(b(s,f)\) computed only for itself. Here, because PF is computed only for itself, all \(\textit{UF}(u)\)s become \(\bar{R}(u)-s\). Therefore, it is computed using only \(\textit{IF}(i)\). \[