variance_nested_two_step

Two-step (nested) variance decomposition. For nested sub-$\sigma$-algebras $m' \le m \le m_0$ and a square-integrable $X$ on $(\Omega, m_0, \mu)$, the total variance splits into three nonnegative pieces:

This is the law of total variance applied twice — once to $X$ with the finer $m$, and once to the martingale $\mathbb{E}[X\mid m]$ with the coarser $m'$ — using the tower property $\mathbb{E}[\mathbb{E}[X\mid m]\mid m'] = \mathbb{E}[X\mid m']$ to identify the residual. It is the inductive heart of the Doob-martingale telescoping that produces the Efron–Stein sum: iterating this nesting over a coordinate filtration $\bot = F_0 \le F_1 \le \dots \le F_n = m_0$ expands $\operatorname{Var}(X)$ as $\sum_k \mathbb{E}[\operatorname{Var}(\mathbb{E}[X\mid F_k]\mid F_{k-1})]$, each term the conditional variance contributed by coordinate $k$.