## 4 Proposed Model
### Observation from Rating Distribution
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**User Type**} & **Rating Range** & **\#Related** & \multirow{2}{*}{**Densities**} & \multicolumn{2}{c}{**Statistical Parameters**} & \multicolumn{2}{c}{**Fitted Parameters**} \\ \cline{5-8}  & **Range** & **Rating** & & **\(\mu_{u}\)** & \(\sigma_{u}\) & \(\mu_{f}\) & \(\sigma_{f}\) \\ \hline \multirow{8}{*}{_Normal_} & [0,1) & 141,610 & 0.0280 & & & & \\ \cline{2-8}  & [1,2) & 322,168 & 0.0638 & & & & \\ \cline{2-8}  & [2,3) & 663,944 & 0.1315 & & & & \\ \cline{2-8}  & [3,4) & 1,719,123 & 0.3405 & & & & \\ \cline{2-8}  & [4,5) & 1,366,972 & 0.2707 & & & & \\ \cline{2-8}  & [5,6) & 835,220 & 0.1654 & & & & \\ \hline \multirow{8}{*}{_Strict_} & [0,1) & 12,135 & 0.1042 & & & & \\ \cline{2-8}  & [1,2) & 15,200 & 0.1305 & & & & \\ \cline{2-8}  & [2,3) & 24,156 & 0.2074 & & & & \\ \cline{2-8}  & [3,4) & 47,324 & 0.4063 & & & & \\ \cline{2-8}  & [4,5) & 17,649 & 0.1515 & & & & \\ \cline{2-8}  & [5,6) & – & 0.0000 & & & & \\ \hline \multirow{8}{*}{_Lenient_} & [0,1) & – & 0.0000 & & & & \\ \cline{2-8}  & [1,2) & 38,980 & 0.0142 & & & & \\ \cline{2-8}  & [2,3) & 156,224 & 0.0570 & & & & \\ \cline{2-8}  & [3,4) & 836,237 & 0.3054 & & & & \\ \cline{2-8}  & [4,5) & 950,718 & 0.3472 & & & & \\ \cline{2-8}  & [5,6) & 756,329 & 0.2762 & & & & \\ \hline \multirow{8}{*}{_Middle_} & [0,1) & – & 0.0000 & & & & \\ \cline{2-8}  & [1,2) & 17,501 & 0.0599 & & & & \\ \cline{2-8}  & [2,3) & 41,515 & 0.1421 & & & & \\ \cline{2-8}  & [3,4) & 138,739 & 0.4750 & & & & \\ \cline{2-8}  & [4,5) & 94,333 & 0.3230 & & & & \\ \cline{2-8}  & [5,6) & – & 0.0000 & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: Statistics of the rating distribution of different kinds of users in Flixster. We demonstrate the related rating densities in stem style in Figure 3 to observe the local distribution of ratings given by different kinds of users. We observed that the rating densities of all kinds of users generally follow normal distributions with different deviations, as shown in stem Figure 3. Intuitively, we fitted these densities into normal distributions based on \(\left(\mu_{u},\ \sigma_{u}\right)\), as shown in the dashed curves in Figure 3. Additionally, for each kind of user, we fitted the complete rating densities into a normal distribution based on the related mean \(\left(\mu_{f},\ \sigma_{f}\right)\), as shown in the blue curves in Figure 3. We found that the statistical distribution of \(N\left(\mu_{u},\ \sigma_{u}{}^{2}\right)\) is close to the fitted \(N\left(\mu_{f},\ \sigma_{f}{}^{2}\right)\). The main differences between them on the same ratings remain in the related standard deviation and related mean value. From the observation, we found that the rating distribution of different kinds of users can also follow the normal distribution. We call this a local distribution in this paper and define that ratings given by a personalized user \(u\) follow the normal distribution \(\varphi_{u}(x)=N\left(\mu_{u},\sigma_{u}{}^{2}\right)\), with the probability density as:  
\[\varphi_{u}(x)=N\left(\mu_{u},\sigma_{u}{}^{2}\right)=\frac{1}{\sqrt{2\pi} \sigma_{u}}e^{-\frac{\left(x-\mu_{u}\right)^{2}}{2\sigma_{u}{}^{2}}}, \tag{5}\]  
where \(x\in\left[\mu_{u}-\frac{Len_{u}}{2},\mu_{u}+\frac{Len_{u}}{2}\right]\) and \(\mu_{u}\) is the local mean value by \(u\) and \(\sigma_{u}\) is the related local standard deviation. (3) Guess and Proposition  
The main purpose of this paper was to find a more rational estimation model based on the global distribution \(\varphi(y)=N\left(\mu,\sigma^{2}\right)\) and local distribution \(\varphi_{u}(x)=N\left(\mu_{u},\sigma_{u}{}^{2}\right)\). Observation proves that different personalized rating ranges result in different standard deviations but still follow local normal distributions. In our opinion, with ratings increasing, the cumulative distribution of different kinds of users' ratings will be closer and the same when there are enough rating behaviors.