RemarkRemark: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Sobol sensitivity and observable-error interface

Remark

Let $1 \leq d < \infty$ , let $(E_{j}, E_{j}, μ_{j})$ be probability spaces, and set $(E, E, μ) = \prod_{j = 1}^{d} (E_{j}, E_{j}, μ_{j})$ with the finite product sigma-algebra and product probability. Let $X = (X_{1}, \dots, X_{d})$ have law $μ$ and let $f \in L^{2} (μ)$ be real-valued. Write $[d] = {1, \dots, d}$ ; coordinate products $E_{v}$ and laws $μ_{v}$ use the one-point probability space for the empty product. For $v \subseteq [d]$ , define the coordinate projection by $P_{v} f (x_{v}) : = \int_{E_{[d] ∖ v}} f (x_{v}, z_{[d] ∖ v}) d μ_{[d] ∖ v} (z_{[d] ∖ v}),$ for almost every integrable section, assigning zero on its exceptional null set, and lift it to $E$ by making it constant in the other coordinates. As $L^{2}$ classes, $P_{[d]} f = f$ and $P_{\emptyset} f = \int f d μ$ . Define the product-law Hoeffding components by $f_{u} : = \sum_{v \subseteq u} (- 1)^{∣ u ∣ - ∣ v ∣} P_{v} f, u \subseteq [d] .$ In particular $f_{\emptyset} = E_{μ} f$ . With these definitions, write $V_{u} : = E [f_{u}^{2}]$ and $V : = Var (f (X)) = \sum_{u \neq \emptyset} V_{u}$ . When $V > 0$ , the first-order and total Sobol indices are $S_{i} : = V_{{i}} / V$ and $S_{T_{i}} : = \sum_{u ∋ i} V_{u} / V$ ; when $V = 0$ , these normalized indices are undefined.

For sample size $n \geq 2$ , draw $n$ IID rows; within each row let $X, X^{'}, U, V_{0}, W, W^{'}$ be mutually independent random vectors with law $μ$ . Define $X^{[i]} = (X_{i}, X_{- i}^{'})$ and ${\tilde{X}}^{[i]} = (X_{- i}, X_{i}^{'})$ . The first-order numerator is $N_{i} : = f (X) f (X^{[i]}) - f (U) f (V_{0})$ , the total-effect numerator is $T_{i} : = \frac{1}{2} (f (X) - f ({\tilde{X}}^{[i]}))^{2}$ , and the denominator is $D : = \frac{1}{2} (f (W) - f (W^{'}))^{2}$ . The expectations of the two numerators are respectively $V_{{i}}$ and $\sum_{u ∋ i} V_{u}$ . Since $D \geq 0$ , its sample mean $\overset{ˉ}{D}$ is nonnegative. If $\overset{ˉ}{D} = 0$ , the estimator returns status zero-empirical-variance, no normalized index, and no plug-in covariance matrix or MCSE. If $\overset{ˉ}{D} > 0$ , it returns ratios of the sample means of $N_{i}$ and $T_{i}$ over $\overset{ˉ}{D}$ , together with the formal plug-in covariance matrix $\hat{J} \hat{Σ} {\hat{J}}^{T} / n$ , where $\hat{Σ} : = (n - 1)^{- 1} \sum_{r = 1}^{n} (Z_{r} - \overset{ˉ}{Z})^{T} (Z_{r} - \overset{ˉ}{Z})$ , $Z_{r} = (N_{1, r}, \dots, N_{d, r}, T_{1, r}, \dots, T_{d, r}, D_{r})$ , and, on this positive-denominator branch, $\hat{J}$ is the derivative of the vector ratio map at the observed row means. The zero-denominator branch can have positive probability even when the population variance $V$ is positive.

A finite-sample ratio need not be unbiased: for one input $X \sim Bernoulli (1 / 2)$ and output $f (x) = x$ , take two rows $r = 1, 2$ of mutually independent Bernoulli $(1 / 2)$ draws $X_{r}, X_{r}^{'}, U_{r}, V_{r}, W_{r}, W_{r}^{'}$ , with all draws independent across rows. Set $N_{r} : = X_{r}^{2} - U_{r} V_{r}$ , $D_{r} : = \frac{1}{2} (W_{r} - W_{r}^{'})^{2}$ , $\overset{ˉ}{N} : = (N_{1} + N_{2}) / 2$ , and $\overset{ˉ}{D} : = (D_{1} + D_{2}) / 2$ . Here $V = Var (X) = 1 / 4$ and $V_{{1}} = E [(X - E X)^{2}] = 1 / 4$ , so the true first-order index is $1$ , while $P (\overset{ˉ}{D} = 0) = 1 / 4$ and conditional on $\overset{ˉ}{D} > 0$ the reported ratio has expectation $E [\overset{ˉ}{N} / \overset{ˉ}{D} ∣ \overset{ˉ}{D} > 0] = 5 / 6$ .

The joint ratio CLT and plug-in MCSE statement applies under its additional assumptions of AC, $f \in L^{4}$ , and $V > 0$ , using any fixed finite extension on the zero-status event only for the asymptotic statements; an $L^{2}$ input alone does not ensure a finite row-vector covariance or an MCSE interpretation. These independent-input formulas are not asserted for dependent coordinates. The sampling and bias budget above bounds an expected scalar observable only; it gives no error bound for Sobol ratios.

Remark references

Product-coordinate averages, empty-product convention and representative interpretation

Hoeffding components defined by the subset sum

First-order and total Sobol index definition

Pick-freeze covariance identities for numerator expectations

Pick-freeze Sobol estimator and its finite-sample qualifications

Sobol ratio CLT and MCSE hypotheses

Axiom of Choice used by the ratio-CLT supplier

Art B. Owen, Variance Components and Generalized Sobol' Indices, arXiv:1205.1774, §§2–2.1

Olivier Le Maître and Omar Knio, Introduction to Uncertainty Quantification, pp. 66–83

Facts & Assumptions

Given: A finite product input law with $f \in L^{2}$ , the row sampling convention in the Statement, and, for the finite-sample witness, a Bernoulli $(1 / 2)$ input with $f (x) = x$ and two independent rows.

[A1]

The input law is a finite product probability law, $f \in L^{2}$ , and the independent copies in the row sampler have that law.

[A2]

For the ratio CLT/MCSE interface only, additionally assume AC, $f \in L^{4}$ , and $V > 0$ .

[F1]

First-order and total Sobol indices: $V_{u} = E [f_{u}^{2}]$ , $V = Var (f) = \sum_{u \neq \emptyset} V_{u}$ , and for $V > 0$ , $S_{1} = V_{{1}} / V$ .

[F2]

Product-law Hoeffding ANOVA components: $f_{{1}} = P_{{1}} f - P_{\emptyset} f$ and $P_{\emptyset} f = E f$ .

[F3]

Moments, variance, and covariance on a probability space: for square-integrable $X$ , $Var (X) = E [(X - E [X])^{2}]$ .

[F4]

Pick-freeze Monte Carlo estimators for Sobol indices: for one input its first-order numerator is $N = f (X) f (X) - f (U) f (V)$ , its denominator row is $D = \frac{1}{2} (f (W) - f (W^{'}))^{2}$ , and independent rows are sampled. On a zero denominator mean it returns zero-empirical-variance with no normalized index or plug-in matrix; on a positive denominator mean it returns the ratios and the formal derivative-based plug-in matrix.

[F5]

Product-coordinate averaging operators: $P_{[d]} f = f$ and $P_{\emptyset} f = \int f d μ$ .

[F6]

Pick-freeze covariances recover first and total Sobol numerators: for a finite product law and $f \in L^{2}$ , the first-order hybrid covariance is $V_{{i}}$ and the half squared total-effect hybrid difference has expectation $\sum_{u ∋ i} V_{u}$ .

[F7]

Pick-freeze Sobol ratios are consistent and have a joint delta-method CLT: under AC, $f \in L^{4}$ , and $V > 0$ , the specified IID row ratios have a joint CLT and the scaled plug-in covariance converges to its stated limit.

Proof

1.1F1F2F3F5givenalgebra

On the one-coordinate product space, [F5] gives $P_{{1}} f = f$ and $P_{\emptyset} f = E [X] = 1 / 2$ . Thus [F2] gives $f_{{1}} = X - 1 / 2$ , and [F3] gives $V_{{1}} = E [(X - 1 / 2)^{2}] = 1 / 4 = V$ . Since $V > 0$ , [F1] gives the true first-order index $S_{1} = 1$ .

1.2F4givenalgebra

In each row, [F4] and $f (x) = x$ give $N_{r} = X_{r} - U_{r} V_{r}$ and $D_{r} = \frac{1}{2} (W_{r} - W_{r}^{'})^{2}$ . Since $U_{r}, V_{r}$ are independent Bernoulli $(1 / 2)$ variables, $E [U_{r} V_{r}] = P (U_{r} = V_{r} = 1) = 1 / 4$ , so $E [N_{r}] = 1 / 2 - 1 / 4 = 1 / 4$ . Since $W_{r}, W_{r}^{'}$ are independent Bernoulli $(1 / 2)$ variables, $P (W_{r} \neq W_{r}^{'}) = 1 / 2$ , so $D_{r} = 1 / 2$ with probability $1 / 2$ and is otherwise zero. The numerator draws are independent of the denominator draws, and the two rows are independent. Hence if $K$ counts the positive $D_{r}$ , then $K \sim Binomial (2, 1 / 2)$ and is independent of $N_{1} + N_{2}$ .

1.3A1F6algebra

The finite-product, square-integrability, and independent-copy hypotheses in [A1] match [F6]. Put $m = E_{μ} f$ . The hybrid $X^{[i]}$ has law $μ$ , since its coordinates are independent with the prescribed marginal laws. Every displayed product is integrable by $2 ∣ a b ∣ \leq a^{2} + b^{2}$ . Expanding the centered covariance in [F6] therefore gives $E [f (X) f (X^{[i]})] = V_{{i}} + m^{2}$ . Independence of $U, V_{0}$ gives $E [f (U) f (V_{0})] = m^{2}$ , so subtraction yields $E N_{i} = V_{{i}}$ . The second identity of [F6] directly gives $E T_{i} = \sum_{u ∋ i} V_{u}$ . These identities do not assert that the normalized sample ratios are unbiased.

2.1F4step 1.2algebra

On the return event $K > 0$ , the sample means are $\overset{ˉ}{D} = K / 4$ and $\overset{ˉ}{N} = (N_{1} + N_{2}) / 2$ . Conditional on $K$ , independence from $N_{1} + N_{2}$ and the mean in step 1.2 give $E [\overset{ˉ}{N} / \overset{ˉ}{D} ∣ K] = 2 E [N_{1} + N_{2}] / K = 1 / K$ .

3.1step 1.1step 1.2step 2.1algebra

Since $P (K = 1) = 1 / 2$ , $P (K = 2) = 1 / 4$ , and $P (K > 0) = 3 / 4$ , the conditional probabilities are $2 / 3$ and $1 / 3$ , and $P (\overset{ˉ}{D} = 0) = P (K = 0) = 1 / 4$ . Therefore $E [\overset{ˉ}{N} / \overset{ˉ}{D} ∣ K > 0] = (2 / 3) 1 + (1 / 3) (1 / 2) = 5 / 6 \neq 1$ . This proves finite-sample bias conditional on the algorithm returning a ratio, while step 1.1 gives the true index.

4.1A2F4F7step 3.1algebra∎

Each $D_{r}$ is nonnegative, so $\overset{ˉ}{D} \geq 0$ . On $\overset{ˉ}{D} = 0$ , [F4] prescribes the failure status and no ratios or plug-in matrix. On $\overset{ˉ}{D} > 0$ , the ratio map is smooth on the open half-space with positive last coordinate; hence its derivative at $\overset{ˉ}{Z}$ and the formal plug-in matrix are defined. The witness in step 3.1 shows that the zero branch need not be a null event even for $V > 0$ . Under the additional hypotheses [A2], [F7] supplies the joint ratio CLT and plug-in MCSE limit, with a fixed finite extension on the zero-status event only for asymptotic statements; that extension does not change what the finite-sample algorithm returns. Those hypotheses are not implied by $f \in L^{2}$ , so no such MCSE interpretation is asserted from square-integrability alone.

Sobol sensitivity and observable-error interface

Remark

Remark references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources