Shapley Information Gain (SIG) Overview

• Shapley Information Gain (SIG) is a family of methods that quantify the marginal information contribution of features or coalitions using information-centric objectives such as predictive entropy, pseudo-R², fact utility, or posterior uncertainty.
• SIG applications span uncertainty explanation in models, regression interpretability through Kullback–Leibler-based pseudo-R², reinforcement learning in proactive healthcare, and Bayesian experimental design for adaptive coalition evaluation.
• Practical implementations of SIG leverage techniques like coalition sampling, brute-force enumeration, and Gaussian process surrogates to overcome combinatorial challenges and provide efficient, interpretable attributions.

Shapley Information Gain (SIG) denotes a family of Shapley-value-based constructions in which marginal contributions are expressed through explicitly information-centered objectives. In "Explaining Predictive Uncertainty with Information Theoretic Shapley Values" (Watson et al., 2023), SIG extends the Shapley framework to explain predictive uncertainty by quantifying each feature's contribution to the conditional entropy of individual model outputs. In "Variable Importance in Generalized Linear Models -- A Unifying View Using Shapley Values" (Acemoglu et al., 2 Jan 2026), SIG is the Shapley decomposition of a Kullback–Leibler-based pseudo-$R^2$. In "ProMed: Shapley Information Gain Guided Reinforcement Learning for Proactive Medical LLMs" (Ding et al., 19 Aug 2025), SIG is an importance-weighted gain in fact coverage used as a reinforcement-learning reward. In "ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation" (Rundel et al., 1 Jun 2026), SIG is the expected information gain about the Shapley vector itself under a Gaussian-process surrogate. The shared structure is coalition-based attribution, but the object being valued differs substantially across these settings.

1. Conceptual scope

The literature uses the label “Shapley Information Gain” for several related but non-identical constructions. Each construction defines a characteristic function over subsets and then applies Shapley averaging, but the underlying target may be predictive entropy, pseudo-$R^2$, fact-level answer utility, or posterior uncertainty about Shapley values.

Setting	Characteristic quantity	SIG object
Predictive uncertainty	$v_{IG}(S,x):=-H(Y\mid X_S=x_S)$	Feature attribution to uncertainty
Generalized linear models	$v(S)=R^2_{\mathrm{KL}(S)}$	Covariate importance
ProMed	$v(S)=\log P_\theta(A^*\mid Q,S)$	Reward for asking questions
ShaplEIG	$I(\phi;y_z\mid D_t)$	Acquisition criterion for coalition evaluation

This plurality matters technically. In the uncertainty setting, SIG explains why a model is uncertain at a specific $x$. In the GLM setting, it decomposes model fit into relative and absolute importance. In ProMed, it weights newly acquired atomic facts by their contextual importance. In ShaplEIG, it is not a post hoc attribution of an existing prediction, but a Bayesian experimental-design criterion for deciding which coalition to evaluate next. A common misconception is therefore to treat SIG as a single universally fixed formula. The published usages instead define a family of Shapley-based information-gain functionals whose semantics depend on the value function.

2. Predictive-uncertainty SIG

For predictive uncertainty, let $X\in\mathbb{R}^d$ be the feature vector, $Y$ the model output whose uncertainty is to be explained, and fix a test point $x$. For any subset $R^2$0, the payoff is defined as

$R^2$1

where

$R^2$2

Equivalent formulations use $R^2$3, or the KL- or cross-entropy games $R^2$4; these differ by $R^2$5-independent constants and yield identical Shapley values (Watson et al., 2023).

With $R^2$6, the attribution assigned to feature $R^2$7 at $R^2$8 is

$R^2$9

Under $v_{IG}(S,x):=-H(Y\mid X_S=x_S)$0, each marginal term is an entropy reduction: $v_{IG}(S,x):=-H(Y\mid X_S=x_S)$1 Because $v_{IG}(S,x):=-H(Y\mid X_S=x_S)$2, each marginal Shapley increment equals

$v_{IG}(S,x):=-H(Y\mid X_S=x_S)$3

Thus the local attribution is an average conditional mutual information over coalitions.

Theoretical properties are explicit. Efficiency yields

$v_{IG}(S,x):=-H(Y\mid X_S=x_S)$4

and under $v_{IG}(S,x):=-H(Y\mid X_S=x_S)$5,

$v_{IG}(S,x):=-H(Y\mid X_S=x_S)$6

the local mutual information. The conditional-independence characterization states that for each $v_{IG}(S,x):=-H(Y\mid X_S=x_S)$7 and any $v_{IG}(S,x):=-H(Y\mid X_S=x_S)$8, $v_{IG}(S,x):=-H(Y\mid X_S=x_S)$9 if and only if $v(S)=R^2_{\mathrm{KL}(S)}$0. Context-specific independence implies $v(S)=R^2_{\mathrm{KL}(S)}$1 at that $v(S)=R^2_{\mathrm{KL}(S)}$2, and the set of distributions for which $v(S)=R^2_{\mathrm{KL}(S)}$3 yet $v(S)=R^2_{\mathrm{KL}(S)}$4 fails has Lebesgue measure zero; the paper describes these as “conspiratorial” exceptions requiring exact cancellation of positive and negative log-ratios. SIG-Shapley also satisfies efficiency, symmetry, sensitivity, and linearity.

The finite-sample inference result uses split conformal bounds. With $v(S)=R^2_{\mathrm{KL}(S)}$5 i.i.d. samples, $v(S)=R^2_{\mathrm{KL}(S)}$6 is used for model fitting and $v(S)=R^2_{\mathrm{KL}(S)}$7 of size $v(S)=R^2_{\mathrm{KL}(S)}$8 for attribution. For target level $v(S)=R^2_{\mathrm{KL}(S)}$9, let $v(S)=\log P_\theta(A^*\mid Q,S)$0 be the $v(S)=\log P_\theta(A^*\mid Q,S)$1-th order statistic of $v(S)=\log P_\theta(A^*\mid Q,S)$2 and $v(S)=\log P_\theta(A^*\mid Q,S)$3 the $v(S)=\log P_\theta(A^*\mid Q,S)$4-th. Then for a new $v(S)=\log P_\theta(A^*\mid Q,S)$5,

$v(S)=\log P_\theta(A^*\mid Q,S)$6

with exactness up to $v(S)=\log P_\theta(A^*\mid Q,S)$7 when the joint of $v(S)=\log P_\theta(A^*\mid Q,S)$8's is continuous. This provides finite-sample, marginal coverage for testing whether $v(S)=\log P_\theta(A^*\mid Q,S)$9 differs from zero.

Practical approximation replaces the $I(\phi;y_z\mid D_t)$0 subset sum by coalition sampling. The method chooses an entropy or variance estimator $I(\phi;y_z\mid D_t)$1, samples out-of-coalition features $I(\phi;y_z\mid D_t)$2 via a model-agnostic kernel, and samples $I(\phi;y_z\mid D_t)$3 coalitions from the Shapley weight $I(\phi;y_z\mid D_t)$4, often with paired sampling and linear-model variance reduction as in KernelSHAP. Williamson & Feng (2020) show that $I(\phi;y_z\mid D_t)$5 coalitions suffices for asymptotically optimal inference. The overall complexity is

$I(\phi;y_z\mid D_t)$6

and in practice $I(\phi;y_z\mid D_t)$7 yields accurate attributions for $I(\phi;y_z\mid D_t)$8 up to a few dozen.

The reported experiments span MNIST digit classification, IMDB sentiment analysis, covariate-shift detection on UCI tabular data, and active feature-value acquisition on a modified Friedman benchmark. In the semi-synthetic missing-data protocol, ranking features by $I(\phi;y_z\mid D_t)$9 yields $x$0 even with $x$1 missingness. Taken together, the method explains not just point predictions but full predictive uncertainty.

3. KL-$x$2 SIG in generalized linear models

In generalized linear models, SIG is defined by decomposing a Kullback–Leibler-based pseudo-$x$3 with Shapley values (Acemoglu et al., 2 Jan 2026). Let $x$4 be observed responses and $x$5 a regular one-parameter exponential family density,

$x$6

The per-observation Kullback–Leibler divergence between $x$7 and $x$8 is

$x$9

With a constant-only null model fitted mean $X\in\mathbb{R}^d$0 and a model fitted mean $X\in\mathbb{R}^d$1, the KL $X\in\mathbb{R}^d$2 is

$X\in\mathbb{R}^d$3

Key special cases are explicit: linear regression gives the classical $X\in\mathbb{R}^d$4, binary logit gives McFadden’s $X\in\mathbb{R}^d$5, and Poisson gives deviance-based $X\in\mathbb{R}^d$6.

Let $X\in\mathbb{R}^d$7 be the covariate indices and $X\in\mathbb{R}^d$8 a coalition. The characteristic function is

$X\in\mathbb{R}^d$9

The resulting SIG for covariate $Y$0 is

$Y$1

The paper emphasizes both relative and absolute importance. Classical Shapley axioms give efficiency, symmetry, dummy, and additivity. For regression interpretability, the additional requirements are monotonicity, the lower bound $Y$2, and the upper bound $Y$3 for the saturated model $Y$4. Because the KL $Y$5 satisfies these, $Y$6 and $Y$7. Relative importance is

$Y$8

and absolute importance is $Y$9.

Closed-form increments are given for linear, logistic, and Poisson regression. In linear regression this recovers Lindeman–Merenda–Gold. In logistic regression the increment is written in terms of McFadden’s $x$0. In Poisson regression the unit deviance is

$x$1

and $x$2 is the explained fraction of total deviance relative to the null model.

Exact computation requires summing over all $x$3 subsets for each $x$4. The paper gives two practical strategies: brute-force for $x$5, and permutation approximation for larger $x$6 by averaging marginal contributions along $x$7 random forward-selection paths. The reported examples include a Poisson doctor-visits model with full-model $x$8, where $x$9; a Poisson-hurdle insurance-claims model with separate SIG decompositions for the two parts; and a geometric-regression doctor-visits model with full $R^2$00, where $R^2$01. These examples illustrate that the framework is designed to interpret both fitted-model importance and importance relative to the saturated-model benchmark.

4. SIG-guided reinforcement learning in proactive medical LLMs

In ProMed, SIG is a reward for interactive medical questioning rather than a post hoc explanation of a conventional predictor (Ding et al., 19 Aug 2025). At dialogue turn $R^2$02, after question $R^2$03 and response $R^2$04, the model’s understanding is $R^2$05. Let $R^2$06 be the full set of atomic facts. The raw information gain of $R^2$07 is

$R^2$08

Fact importance is then defined with a Shapley value over atomic facts. For any subset $R^2$09,

$R^2$10

and the Shapley value of fact $R^2$11 is $R^2$12. After softmax normalization,

$R^2$13

the SIG reward becomes

$R^2$14

The intended decomposition is explicit: quantity is measured by newly entailed facts, while contextual importance is measured by Shapley values that encode standalone utility and synergy or competition with other facts.

The computation of $R^2$15 uses a Monte Carlo approximation over random permutations, with maximum iterations $R^2$16 and tolerance $R^2$17. After each question, the system elicits current understanding $R^2$18 via a “doctor understanding” prompt, runs a fact-checker on each $R^2$19, and computes $R^2$20.

ProMed integrates SIG into a two-stage training pipeline. Stage 1, SIG-Guided Model Initialization, runs Monte Carlo Tree Search from partial input $R^2$21, scores question nodes with $R^2$22, retains the best answer-correct trajectory per case, and fine-tunes via supervised loss. A complete trajectory

$R^2$23

receives

$R^2$24

Stage 2, SIG-Augmented Policy Optimization, builds on Group Relative Policy Optimization and decomposes trajectory reward into action-level signals. Question $R^2$25 receives

$R^2$26

while the final answer receives

$R^2$27

The case study in rheumatology centers on the missing key fact nail pitting. After training, the model asks, “Do you have any nail changes (pitting, onycholysis)?”, the fact-checker finds that $R^2$28 newly appears in $R^2$29, and if $R^2$30 and no other facts are gained, then $R^2$31. Experimentally, across three LLMs and two benchmarks, ProMed(Stage 1+2) achieves up to $R^2$32 vs. second-best $R^2$33 on MedQA and $R^2$34 vs. $R^2$35 on CMB, with average relative improvement over second-best $R^2$36 and a $R^2$37 gain over the direct baseline. Ablations that remove SIG or its components degrade performance by up to $R^2$38 points.

5. SIG as expected information gain for adaptive coalition selection

ShaplEIG uses SIG as an acquisition criterion for estimating Shapley values when value-function evaluations are expensive (Rundel et al., 1 Jun 2026). Let $R^2$39 be the player set, $R^2$40 the expensive value of coalition $R^2$41, and $R^2$42 the vector of Shapley values for $R^2$43. At iteration $R^2$44, with data

$R^2$45

where each $R^2$46 encodes a coalition and $R^2$47, $R^2$48, the next coalition is chosen to maximize

$R^2$49

Here the quantity is the expected reduction in differential entropy about the Shapley vector after observing $R^2$50.

A Gaussian-process prior is placed on $R^2$51 over $R^2$52, with zero mean and a Hamming-distance kernel. Writing $R^2$53 for all coalitions, the posterior is

$R^2$54

By linearity of Shapley values,

$R^2$55

where $R^2$56 is the fixed matrix of Shapley weights. Hence

$R^2$57

Under this linear-Gaussian structure, mutual information is available in closed form. For a one-point design $R^2$58 with observation $R^2$59, SIG simplifies to

$R^2$60

where $R^2$61 and $R^2$62. The paper states that $R^2$63 is the marginal posterior variance at $R^2$64, and $R^2$65 reflects the variance after conditioning on $R^2$66.

Naïve evaluation is exponential in $R^2$67, because it would require manipulating $R^2$68 matrices. ShaplEIG reduces this to polynomial complexity by exploiting the product structure of the Hamming kernel and the combinatorial structure of $R^2$69. Elementary symmetric polynomials are used for the linear term $R^2$70, and bivariate generating polynomials with pre- and suffix table convolutions are used for the quadratic term $R^2$71. The resulting complexity is $R^2$72 for a single candidate and $R^2$73 for a batch of $R^2$74 candidates.

The practical loop alternates GP hyperparameter fitting, SIG evaluation over candidate coalitions, and selection of $R^2$75. After $R^2$76 costly evaluations, the posterior mean $R^2$77 yields consistent Shapley-value estimates. Empirically, across nine real-world costly games with $R^2$78 players, ShaplEIG outperforms or matches stochastic, surrogate-based, and fixed-design Bayesian experimental-design baselines, with gains largest in the low-budget regime $R^2$79.

6. Common structure, distinctions, and recurrent themes

Across these formulations, SIG is always built from subset-based evaluation and Shapley aggregation, but the meaning of “information gain” changes with the task. In predictive uncertainty, it is a reduction in local conditional entropy and is directly identified with conditional mutual information (Watson et al., 2023). In generalized linear models, it is the Shapley decomposition of increments in $R^2$80, a goodness-of-fit quantity normalized to $R^2$81 (Acemoglu et al., 2 Jan 2026). In ProMed, it is the importance-weighted gain in entailed atomic facts that guides question asking (Ding et al., 19 Aug 2025). In ShaplEIG, it is mutual information between a future coalition observation and the Shapley vector under a GP posterior (Rundel et al., 1 Jun 2026).

This suggests a useful unifying view: SIG is less a single estimator than a design pattern in which a coalition value function is chosen to represent uncertainty reduction, goodness-of-fit gain, clinical fact utility, or posterior learning about attributions. The design choice that determines the interpretation is the characteristic function $R^2$82, not the Shapley operator itself.

Several recurrent technical themes also appear across the literature. Exact computation is combinatorial or exponential unless additional structure is exploited. The uncertainty paper uses coalition sampling from the Shapley weight and model-agnostic reference distributions; the GLM paper uses brute-force or permutation approximation; ProMed uses Monte Carlo permutations over atomic facts; ShaplEIG obtains polynomial-time evaluation through Gaussian-process linearity and elementary symmetric polynomials. Another recurrent theme is that SIG is not restricted to explanation in the narrow sense. The published applications include covariate-shift detection, active learning, feature selection, active feature-value acquisition, proactive medical dialogue, data valuation, hyperparameter importance, and local explanations.

The term therefore carries both continuity and ambiguity. The continuity lies in Shapley-based averaging of marginal gains under a carefully chosen information-centric objective. The ambiguity lies in the fact that different papers use “SIG” for substantively different targets. For technical reading, the decisive question is always which random variable, fit criterion, fact set, or posterior uncertainty the information gain is defined over.