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A pick-freeze ratio can be biased at finite sample size

Example

Take one input $X \sim Bernoulli (1 / 2)$ and $f (X) = X$ . The true first-order Sobol index is $1$ . With two rows of Pick-freeze Monte Carlo estimators for Sobol indices, conditional on the algorithm returning an index (its empirical variance denominator is positive), the expected reported ratio is $5 / 6$ . Thus unbiased numerator and denominator estimators do not make their ratio unbiased.

Facts & Assumptions

Given: The one-factor Bernoulli model and two independent rows of the stated pick-freeze bundle.

[F1]

Pick-freeze Monte Carlo estimators for Sobol indices: the one-factor numerator is $N = f (X) f (X) - f (U) f (V)$ and the independent-pair denominator is $D = \frac{1}{2} (f (W) - f (W^{'}))^{2}$ ; the ratio $\overset{ˉ}{N} / \overset{ˉ}{D}$ is returned when the sample mean $\overset{ˉ}{D} > 0$ .

[F2]

Pick-freeze covariances recover first and total Sobol numerators: the population first-order numerator and variance denominator are both $V_{{1}}$ and $V$ .

[F3]

First-order and total Sobol indices: $S_{1} = V_{{1}} / V$ when $V > 0$ .

Verification

Given: The model and algorithm convention in the Example.

1.1F2F3givenalgebra

Since $E X = 1 / 2$ and $Var (X) = 1 / 4$ , the only ANOVA component is $f_{{1}} = X - 1 / 2$ , so $V_{{1}} = V = 1 / 4$ and [F3] gives the true index $S_{1} = 1$ .

1.2F1givenalgebra

In each row $r$ , the first-order hybrid equals $X_{r}$ , so $N_{r} = X_{r} - U_{r} V_{r}$ and $E N_{r} = 1 / 4$ . Also $D_{r} = 1 / 2$ exactly when $W_{r} \neq W_{r}^{'}$ , an event of probability $1 / 2$ , and is zero otherwise. All numerator draws are independent of all denominator draws. If $K$ is the number of positive denominators in the two rows, then $K \sim Binomial (2, 1 / 2)$ and is independent of $N_{1} + N_{2}$ .

2.1F1F2step 1.1step 1.2algebra∎

On the return event $K > 0$ , $\overset{ˉ}{D} = K / 4$ and $\overset{ˉ}{N} = (N_{1} + N_{2}) / 2$ . Independence and $E [N_{1} + N_{2}] = 1 / 2$ give $E [\overset{ˉ}{N} / \overset{ˉ}{D} ∣ K] = 1 / K$ . Since $P (K = 1 ∣ K > 0) = 2 / 3$ and $P (K = 2 ∣ K > 0) = 1 / 3$ , the conditional reported-ratio expectation is $2 / 3 + (1 / 3) (1 / 2) = 5 / 6$ , different from the population index $1$ in step 1.1. Each $N_{r}$ and $D_{r}$ remains unbiased for $1 / 4$ , but their ratio is biased when returned.

Depends on

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Sources

Art B. Owen, Variance Components and Generalized Sobol' Indices, §2.1 (standard reference, not scraped)