shapiq: Shapley Interactions for Machine Learning

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Möbius Interactions

$$ \Phin(S) := \sum{T \subseteq S} (-1)^{|S|-|T|}\nu(T) $$

Where:

Symbol	Description
$\Phi_n(S)$	Möbius Interaction for subset S
$S$	A subset of players
$T$	A subset of S
$n$	Total number of players (features)
$\nu(T)$	Value function of the cooperative game for subset T

Recovery Property

The recovery property related to MIs is defined as:

$$
\nu(T) = \sum_{S \subseteq T} \Phi_n(S) $$

Where:	Symbol	Description
$\nu(T)$	Value function of the cooperative game for subset T
$S$	A subset of T
$\Phi_n(S)$	Möbius Interaction for subset S
$T$	A subset of players
$n$	Total number of players

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SHAPIQ utilizes MIs as an additional method for analyzing feature contributions

Feature	Möbius Interactions (MIs)	SHAP
Contribution Calculation	Considers the interactions of all feature combinations and calculates contributions of higher-order interactions.	Calculates the marginal contribution of each feature in all possible combinations.		Explanation Depth	Provides detailed explanations of higher-order feature interactions, aiding in understanding the relationships between features.	Provides individual contribution values for features, intuitively showing the impact of each feature on predictions.
Computational Complexity	May be more complex in calculations, especially when involving multiple higher-order interactions.	While computationally complex, typically focuses on simpler feature contribution explanations.
Application Scenarios	Suitable for situations that require in-depth analysis of feature interactions.	Suitable for situations that need quick retrieval of each feature's contribution.
Tool Support	Introduces the shapiq Python package for efficient computation of MIs.	Widely used tools and libraries (such as the SHAP library) are ava

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Experimental Comparisons

The authors utilize a suite of benchmark games that cover different machine learning tasks. They compute both MIs and SHAP values for these benchmarks and evaluate metrics such as accuracy, precision, and computational efficiency. The results show how MIs can provide a clearer understanding of higher-order feature interactions compared to traditional SHAP values.

Index	Description
k-SII	Shapley Interaction Index of order ( k ); accounts for interactions among features up to a specified order.
FSII	Faithful Shapley Interaction Index; ensures interactions are faithful to the underlying model's predictions.
FBII	Faithful Banzhaf Interaction Index; based on the Banzhaf value, aims to provide a faithful representation of interactions among features.
STII	Shapley Taylor Interaction Index; emphasizes top-order interactions, highlighting the most significant combinations of features.