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Conditional variance of a fixed regression-tree partition

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

.

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