# Variable Selection Using Deep Variational Information Bottleneck with Drop-Out-One Loss
## 4 Experimental Analysis
### Simulated Experiments
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Example 1**} & \multicolumn{4}{c}{**Example 2**} \\ \cline{2-9}
**Models** & **TP** & **FP** & **STP** & **TP** & **FP** & **STP** \\ \hline Lasso [1] & 4.35 & 0.65 & 0.85 & 1.30 & 0.70 & 0.45 \\ Elastic Net [32] & 4.55 & 0.45 & 0.80 & 1.50 & 0.50 & 0.50 \\ SCAD [3] & 4.90 & 0.10 & 0.30 & 1.30 & 0.70 & 0.45 \\ RF [13] & 4.50 & 0.50 & 0.74 & **2.00** & **0.00** & **0.00** \\ AdaBoost [33] & 4.75 & 0.25 & 0.62 & 1.90 & 0.10 & 0.30 \\ SpAM [4] & **5.00** & **0.00** & **0.00** & 1.35 & 0.65 & 0.47 \\ GL [7] & 4.80 & 0.20 & 0.50 & 1.20 & 0.800 & 0.40 \\ Knockoff [11] & 4.90 & 0.10 & 0.40 & 1.60 & 0.40 & 0.48 \\ Lasso Net [14] & **5.00** & **0.00** & **0.00** & 1.85 & 0.15 & 0.35 \\ DeepVIB (G) & **5.00** & **0.00** & **0.00** & **2.00** & **0.00** & **0.00** \\ DeepVIB (E) & **5.00** & **0.00** & **0.00** & **2.00** & **0.00** & **0.00** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The averaged performance of variable selection (Examples 1 and 2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**\(\Sigma=\textbf{I}\)**} & \multicolumn{4}{c}{\(\Sigma_{i,j}=\textbf{0.5}^{|i-j|}\)} \\ \cline{2-9}
**(\(N,p\))** & **Models** & **TP** & **FP** & **STP** & **TP** & **FP** & **STP** \\ \hline Lasso [1] & 3.50 & 0.50 & 1.11 & 2.50 & 1.50 & 1.16 \\ Elastic Net [32] & 3.70 & 0.30 & 0.78 & 3.50 & 0.50 & 1.07 \\ SCAD [3] & 3.85 & 0.15 & 0.47 & 3.50 & 0.50 & 1.02 \\ RF [13] & 3.90 & 0.10 & 0.30 & 3.75 & 0.25 & 0.50 \\ AdaBoost [33] & 3.65 & 0.35 & 0.65 & 3.35 & 0.65 & 0.96 \\ SpAM [4] & **4.00** & **0.00** & **0.00** & **4.00** & **0.00** & **0.00** \\ GL [7] & 3.45 & 0.55 & 0.97 & 3.20 & 0.80 & 1.02 \\ Knockoff [11] & 3.70 & 0.30 & 0.95 & 3.55 & 0.45 & 1.11 \\ Lasso Net [14] & **4.00** & **0.00** & **0.00** & **4.00** & **0.00** & **0.00** \\ Lasso Net [14] & **4.00** & **0.00** & **0.00** & **4.00** & **0.00** & **0.00** \\ DeepVIB (G) & **4.00** & **0.00** & **0.00** & **4.00** & **0.00** & **0.00** \\ DeepVIB (E) & **4.00** & **0.00** & **0.00** & **4.00** & **0.00** & **0.00** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The averaged performance of variable selection (Example 3). (1) Table 2 shows that the TPs of the proposed DeepVIB (G) and DeepVIB (E) in Examples 1 and 2 always equal the \(p_{true}\), while the FPs and STPs are always 0. This means that the proposed method can perfectly select all the relevant variables in each experiment. For comparison, other algorithms always select the irrelevant variables in Example 1 or 2. This supports the superiority of DeepVIB for variable selection. (2) Tables 3 and 4 show the performance of all variable-selection algorithms in Examples 3 and 4. Generally, the performance degrades when \(p\) increases or when the input variables become correlated. This is consistent with the previous phenomenon in Examples 1 and 2. In particular, the proposed DeepVIB (G) and DeepVIB (E) can select all the relevant variables in the cases \((N,p)=(100,10)\) and \((100,20)\), which is better performance compared with the other variable-selection algorithms. (3) From the empirical results in Tables 3 and 4, we also note that DeepVIB (G) provides consistently larger TPs compared with DeepVIB (E) in the case \((N,p)=(200,50)\). This indicates that DeepVIB can achieve better variable selection results when \(p(z)\) obeys the Gaussian distribution.