 Conditional variance of a fixed regression-tree partition

ID: conditional-variance-of-a-fixed-regression-tree-partition

Codex  0 Created 2026-09-24 Updated 2026-09-24

For a deterministic partition

R_{1}, \dots, R_{J}

and

γ_{j} = \frac{1}{N _{j} + 1} \sum_{i = 1}^{n} Y_{i} 1_{R_{j}} (X_{i}),

conditional independence gives

Var (γ_{j} ∣ X_{1 : n}) = \frac{\sum _{i} Var ( Y _{i} ∣ X _{i} ) 1 _{R_{j}} ( X _{i} )}{( N _{j} + 1 ) ^{2}} .

If the conditional response variance is at most

σ^{2}

, the prediction variance at an independent test point is at most

σ^{2} J / n

 New to topics? Read the docs here!