Sobol Index Calculation - Use Emulator Uncertainty if available

[H-Sax]

Description

jrsssb_66_3_751.pdf

Current Issue with Mean-Only Approach

The current implementation uses only the GP mean prediction to calculate Sobol indices, but this misses a crucial component of uncertainty. When you only use the mean, you're assuming perfect knowledge of the true function - which defeats the purpose of having a probabilistic emulator.

What's Missing: Emulator Uncertainty

From the paper attached, we should treat the computer model as an unknown function with a GP posterior distribution. This means:

• Sobol indices themselves become random variables (not fixed numbers)
• We need posterior expectations instead of point estimates:

$$\hat{S}_i = \frac{E^*[V_i]}{E^*[\text{var}(Y)]}$$

• As a consequence there is a modification to the standard Hoeffding-Sobol decomposition:

$$E^*[\text{var}\{z_i(X_i)\}] = \text{var}[E^*\{z_i(X_i)\}] + E[\text{var}^*\{z_i(X_i)\}]$$

Defining the Terms:

• $z_i(x_i) = E(Y|x_i) - E(Y)$ is the main effect of input $X_i$
• $E^*[\cdot]$ denotes posterior expectation with respect to the GP
• $\text{var}^*[\cdot]$ denotes posterior variance with respect to the GP
• $V_i = \text{var}{z_i(X_i)}$ is the main effect variance

Variance Decomposition Breakdown:

1. $\text{var}[E^*{z_i(X_i)}]$: Variation of the posterior mean prediction (what current implementation captures)

2. $E[\text{var}^*{z_i(X_i)}]$: Average uncertainty within the GP posterior about what the main effect actually is (currently ignored)

The Problem with Point Estimates

When using only the GP mean, you're effectively setting $E[\text{var}^*{z_i(X_i)}] = 0$ - assuming perfect knowledge and no emulator uncertainty. This can seriously underestimate uncertainty in sensitivity indices.

Reproduction steps

The paper provides the formalisation on how one should properly compute Sobol indices for probablistic emulators.

Version

jrsssb_66_3_751.pdf

Screenshots

![DESCRIPTION](LINK.png)

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