Sobol Indices

Explainability Fairness

Description

Sobol Indices quantify how much each input feature contributes to the total variance in a model's predictions through global sensitivity analysis. The technique calculates first-order indices (individual feature contributions) and total-order indices (including all interaction effects involving that feature). By systematically sampling the input space and decomposing output variance, Sobol Indices reveal which features drive model uncertainty and which interactions between features are most important for predictions.

Example Use Cases

Explainability

Analysing a climate prediction model to determine which atmospheric parameters (temperature, humidity, pressure) contribute most to rainfall forecast uncertainty, helping meteorologists understand which measurements need the highest precision.

Evaluating a financial risk model to identify which economic indicators (interest rates, inflation, GDP growth) drive the most variability in portfolio value predictions, enabling better risk management strategies.

Fairness

Analysing a credit scoring model to quantify how much prediction variance stems from zip code (a potential proxy for race), helping identify features that may cause disparate impact across demographic groups.

Limitations

Computationally expensive, requiring thousands of model evaluations to achieve stable variance estimates, making it impractical for very slow models.
Assumes input features are independently distributed, which can lead to misleading results when features are correlated in real data.
Curse of dimensionality makes the technique increasingly difficult and expensive to apply as the number of input features grows beyond 10-20.
Requires defining appropriate probability distributions for input features, which may not accurately reflect real-world feature distributions.

Resources

Sobol Tensor Trains for Global Sensitivity Analysis

Research Paper•Rafael Ballester-Ripoll, Enrique G. Paredes, and Renato Pajarola•Dec 1, 2017

Sobol indices — UQpy v4.2.0 documentation

Documentation

Sobol Indices to Measure Feature Importance | Towards Data Science

Tutorial

Basics — SALib's documentation

Documentation

UQpy (Uncertainty Quantification with python)

Software Package

SALib/SALib

Software Package

Related Techniques

Name	Description	Assurance Goals
Relabelling	A preprocessing fairness technique that modifies class labels in training data to achieve equal positive outcome rates across protected groups. Also known as 'data massaging', this method identifies instances that contribute to discriminatory patterns and flips their labels (from positive to negative or vice versa) to balance the proportion of positive outcomes between demographic groups. The technique aims to remove historical bias from training data whilst preserving the overall class distribution, enabling standard classifiers to learn from discrimination-free datasets.	Fairness Transparency Reliability
ANCHOR	ANCHOR generates high-precision if-then rules that explain individual predictions by identifying the minimal set of feature conditions that guarantee a specific prediction with high confidence. It searches for 'anchor' conditions (e.g., 'age > 30 AND income < £50k') that ensure the model gives the same prediction at least 95% of the time when those conditions are met. This creates human-readable rules that users can trust as sufficient conditions for understanding why a particular decision was made.	Explainability Transparency
Coefficient Magnitudes (in Linear Models)	Coefficient Magnitudes assess feature influence in linear models by examining the absolute values of their coefficients. Features with larger absolute coefficients are considered to have a stronger impact on the prediction, while the sign of the coefficient indicates the direction of that influence (positive or negative). This technique provides a straightforward and transparent way to understand the direct linear relationship between each input feature and the model's output.	Explainability Transparency
DeepLIFT	DeepLIFT (Deep Learning Important FeaTures) explains neural network predictions by decomposing the difference between the actual output and a reference output back to individual input features. It compares each neuron's activation to a reference activation (typically from a baseline input like all zeros or the dataset mean) and propagates these differences backwards through the network using chain rule modifications. Unlike gradient-based methods, DeepLIFT satisfies the sensitivity property (zero input gets zero attribution) and provides more stable attributions by using discrete differences rather than gradients.	Explainability Transparency
Individual Conditional Expectation Plots	ICE plots display the predicted output for individual instances as a function of a feature, with all other features held fixed for each instance. Each line on an ICE plot represents one instance's prediction trajectory as the feature of interest changes, revealing whether different instances are affected differently by that feature.	Explainability
Integrated Gradients	Integrated Gradients is an attribution technique that explains a model's prediction by quantifying the contribution of each input feature. It works by accumulating gradients along a straight path from a user-defined baseline input (e.g., a black image or an all-zero vector) to the actual input. This path integral ensures that the attributions satisfy fundamental axioms like completeness (attributions sum up to the difference between the prediction and the baseline prediction) and sensitivity (non-zero attributions for features that change the prediction). The output is a set of importance scores, often visualised as heatmaps, indicating which parts of the input were most influential for the model's decision.	Explainability

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