key idea of selective inference

1. What is Selective Inference?
• When you use a feature selection method (e.g. Lasso), there
exists selection bias.
• There are features : 𝑥1, 𝑥2, … 𝑥100, and 𝑥2, 𝑥21, 𝑥30 are chosen.
• Then you make the model : y = 𝑤2 𝑥2 + 𝑤21 𝑥21 + 𝑤30 𝑥30
• Fit it, and get the parameters : 𝑤2 = +3.8, 𝑤21 = −4.5, 𝑤30 = +6.2
• p-values of 𝑤2, w21, w30 are beyond 𝛼 = 0.05.
• However, you cannot directly say that :
“𝑤2 and w30 have positive effects and 𝑤21 has negative effect.”
• Hypothesis selected by some algorithm has selection bias and we
have to correct the bias. To do that, we get following 𝑙 𝑎𝑛𝑑 𝑢.
Ρ 𝑤𝑖 ∉ 𝑙, 𝑢 𝑖 ∈ 𝑆, 𝑆 ← 𝐿𝑎𝑠𝑠𝑜) < 𝛼
• このような条件付分布に基づく統計的推論の枠組をSelective Inference と呼ぶようです