= Conditional variance of a fixed regression-tree partition
For a deterministic partition $R_1,\ldots,R_J$ and
$$
\widetilde\gamma_j=\frac1{N_j+1}
\sum_{i=1}^nY_i\mathbf1_{R_j}(X_i),
$$
conditional independence gives
$$
\operatorname{Var}(\widetilde\gamma_j\mid X_{1:n})
=\frac{\sum_i\operatorname{Var}(Y_i\mid X_i)\mathbf1_{R_j}(X_i)}
{(N_j+1)^2}.
$$
If the conditional response variance is at most $\sigma^2$, the prediction variance at an independent test point is at most $\sigma^2J/n$.
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