---
title: Shapley Information Gain (SIG) Overview
url: https://www.emergentmind.com/topics/shapley-information-gain-sig
type: topic
---

# Shapley Information Gain (SIG) Overview

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" [2306.05724], 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" [2601.00773], 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" [2508.13514], SIG is an importance-weighted gain in fact coverage used as a reinforcement-learning reward. In "ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation" [2606.02247], 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 \(S\subseteq\{1,\dots,d\}\), the payoff is defined as
\[
v_{IG}(S,x):=-H(Y\mid X_S=x_S),
\]
where
\[
H(Y\mid X_S=x_S)=-\int p(y\mid x_S)\log p(y\mid x_S)\,dy.
\]
Equivalent formulations use \(v_H(S,x):=+H(Y\mid X_S=x_S)\), or the KL- or cross-entropy games \(v_{KL},v_{CE}\); these differ by \(S\)-independent constants and yield identical Shapley values [2306.05724].

With \(F=\{1,\dots,d\}\), the attribution assigned to feature \(i\) at \(x\) is
\[
\phi_i(x)=
\sum_{S\subseteq F\setminus\{i\}}
\frac{|S|!\,(d-|S|-1)!}{d!}
\Bigl[v(S\cup\{i\},x)-v(S,x)\Bigr].
\]
Under \(v=v_{IG}\), each marginal term is an entropy reduction:
\[
v_{IG}(S\cup\{i\},x)-v_{IG}(S,x)
=
-H(Y\mid X_S=x_S,X_i=x_i)+H(Y\mid X_S=x_S).
\]
Because \(H(Y\mid X_S)-H(Y\mid X_S,X_i)=I(Y;X_i\mid X_S)\), each marginal Shapley increment equals
\[
\Delta_{IG}(S,i,x)=I(Y;X_i\mid X_S=x_S).
\]
Thus the local attribution is an average conditional mutual information over coalitions.

Theoretical properties are explicit. Efficiency yields
\[
\sum_{i=1}^d \phi_i(x)=v([d],x)-v(\varnothing,x),
\]
and under \(v_{IG}\),
\[
\sum_{i=1}^d \phi_i(x)=-H(Y\mid X=x)+H(Y)=I(Y;X=x),
\]
the local mutual information. The conditional-independence characterization states that for each \(v\in\{v_{KL},v_{CE},v_{IG},v_H\}\) and any \(S\subseteq F\setminus\{i\}\), \(Y\perp X_i\mid X_S\) if and only if \(\sup_x |\Delta_v(S,i,x)|=0\). Context-specific independence implies \(\Delta_v(S,i,x)=0\) at that \(x\), and the set of distributions for which \(\Delta_v(S,i,x)=0\) yet \(Y\perp X_i\mid X_S=x_S\) 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 \(n\) i.i.d. samples, \(I_1\) is used for model fitting and \(I_2\) of size \(m=n/2\) for attribution. For target level \(\alpha\), let \(\hat q_{lo}\) be the \(\lceil (m+1)(\alpha/2)\rceil\)-th order statistic of \(\{\phi_i(x^{(j)}):j\in I_2\}\) and \(\hat q_{hi}\) the \(\lceil (m+1)(1-\alpha/2)\rceil\)-th. Then for a new \(x\),
\[
\Pr\bigl(\phi_i(x)\in[\hat q_{lo},\hat q_{hi}]\bigr)\ge 1-\alpha,
\]
with exactness up to \(2/(n+2)\) when the joint of \(\phi\)'s is continuous. This provides finite-sample, marginal coverage for testing whether \(\phi_i(x)\) differs from zero.

Practical approximation replaces the \(2^d\) subset sum by coalition sampling. The method chooses an entropy or variance estimator \(h(x)\), samples out-of-coalition features \(X_{\neg S}\) via a model-agnostic kernel, and samples \(M\ll 2^d\) coalitions from the Shapley weight \(\Pr(|S|=k)\propto \binom{d-1}{k}^{-1}\), often with paired sampling and linear-model variance reduction as in KernelSHAP. Williamson & Feng (2020) show that \(M=\Theta(n)\) coalitions suffices for asymptotically optimal inference. The overall complexity is
\[
O(M\cdot C_h + M\cdot C_{ref} + M\cdot(\text{cost of }h\text{ evaluation})),
\]
and in practice \(M\sim 10^3\!-\!10^4\) yields accurate attributions for \(d\) 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 \(|\phi_i|\) yields \(\mathrm{ROC\ AUC}\approx 0.68\) even with \(50\%\) missingness. Taken together, the method explains not just point predictions but full predictive uncertainty.

## 3. KL-\(R^2\) SIG in generalized linear models

In generalized linear models, SIG is defined by decomposing a Kullback–Leibler-based pseudo-\(R^2\) with Shapley values [2601.00773]. Let \(y_1,\dots,y_n\) be observed responses and \(f(y;\theta)\) a regular one-parameter exponential family density,
\[
f(y;\theta)=\exp\{y\theta-b(\theta)+c(y)\}.
\]
The per-observation Kullback–Leibler divergence between \(\theta_1\) and \(\theta_2\) is
\[
K(\theta_1,\theta_2)
=
2\,E_{\theta_1}\!\left[\log\frac{f(y;\theta_1)}{f(y;\theta_2)}\right]
=
2\Bigl[(\theta_1-\theta_2)\mu(\theta_1)-\bigl(b(\theta_1)-b(\theta_2)\bigr)\Bigr].
\]

With a constant-only null model fitted mean \(\hat\mu_0\) and a model fitted mean \(\hat\mu\), the KL \(R^2\) is
\[
R^2_{\mathrm{KL}}
=
1-\frac{K(\mathbf y,\hat\mu)}{K(\mathbf y,\hat\mu_0)}
=
1-\frac{\sum_i[\ell(y_i,y_i)-\ell(y_i,\hat\mu_i)]}
{\sum_i[\ell(y_i,y_i)-\ell(y_i,\hat\mu_{0,i})]}
\in[0,1].
\]
Key special cases are explicit: linear regression gives the classical \(R^2\), binary logit gives McFadden’s \(R^2\), and Poisson gives deviance-based \(R^2\).

Let \(P=\{1,\dots,p\}\) be the covariate indices and \(S\subseteq P\) a coalition. The characteristic function is
\[
v(S)=R^2_{\mathrm{KL}(S)}=
1-\frac{K(\mathbf y,\hat\mu_S)}{K(\mathbf y,\hat\mu_0)}.
\]
The resulting SIG for covariate \(j\) is
\[
\mathrm{SIG}_j
=
\sum_{S\subseteq P\setminus\{j\}}
\frac{|S|!\,(p-|S|-1)!}{p!}
\Bigl[
R^2_{\mathrm{KL}(S\cup\{j\})}
-R^2_{\mathrm{KL}(S)}
\Bigr].
\]

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 \(v(\varnothing)=0\), and the upper bound \(v(P')=1\) for the saturated model \(P'\). Because the KL \(R^2\) satisfies these, \(\phi_j\ge 0\) and \(\sum_j \phi_j=R^2_{\mathrm{KL}(P)}\). Relative importance is
\[
\mathrm{imp}^{\mathrm{fit}}_j=\frac{\phi_j}{R^2_{\mathrm{KL}(P)}},
\]
and absolute importance is \(\mathrm{imp}^{\mathrm{best}}_j=\phi_j\in[0,1]\).

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 \(R^2\). In Poisson regression the unit deviance is
\[
d(y_i,\mu_i)=2\Bigl[y_i\ln\frac{y_i}{\mu_i}-(y_i-\mu_i)\Bigr],
\]
and \(R^2_{\mathrm{KL}(S)}\) is the explained fraction of total deviance relative to the null model.

Exact computation requires summing over all \(2^{p-1}\) subsets for each \(j\). The paper gives two practical strategies: brute-force for \(p\lesssim 20\), and permutation approximation for larger \(p\) by averaging marginal contributions along \(M\) random forward-selection paths. The reported examples include a Poisson doctor-visits model with full-model \(R^2_{\mathrm{KL}}=0.2211\), where \(\mathrm{SIG}_{\mathrm{reduced}}=0.1250\); a Poisson-hurdle insurance-claims model with separate SIG decompositions for the two parts; and a geometric-regression doctor-visits model with full \(R^2_{\mathrm{KL}}=0.0953\), where \(\mathrm{SIG}_{\mathrm{chronic}}=0.0535\). 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 [2508.13514]. At dialogue turn \(t\), after question \(q_t\) and response \(r_t\), the model’s understanding is \(U_t\). Let \(\mathcal F=\{f_1,\dots,f_n\}\) be the full set of atomic facts. The raw information gain of \(q_t\) is
\[
\mathrm{IG}(q_t)=\frac{1}{|\mathcal F|}\sum_{i=1}^n
\Bigl[\mathbf 1(f_i\subseteq U_t)-\mathbf 1(f_i\subseteq U_{t-1})\Bigr].
\]

Fact importance is then defined with a Shapley value over atomic facts. For any subset \(S\subseteq\mathcal F\),
\[
v(S)=\log P_\theta(A^*\mid Q,S),
\]
and the Shapley value of fact \(f_i\) is \(\phi(f_i)\). After softmax normalization,
\[
\tilde\phi_i=\frac{\exp(\phi(f_i))}{\sum_{j=1}^n\exp(\phi(f_j))},
\]
the SIG reward becomes
\[
\mathrm{SIG}(q_t)=
\sum_{i=1}^n \tilde\phi_i
\Bigl[\mathbf 1(f_i\subseteq U_t)-\mathbf 1(f_i\subseteq U_{t-1})\Bigr].
\]
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 \(\phi(f_i)\) uses a Monte Carlo approximation over random permutations, with maximum iterations \(K\) and tolerance \(\varepsilon\). After each question, the system elicits current understanding \(U_t\) via a “doctor understanding” prompt, runs a fact-checker on each \(f_i\), and computes \(\mathrm{SIG}(q_t)\).

ProMed integrates SIG into a two-stage training pipeline. Stage 1, SIG-Guided Model Initialization, runs Monte Carlo Tree Search from partial input \(Q_p\), scores question nodes with \(\mathrm{SIG}(q_t)\), retains the best answer-correct trajectory per case, and fine-tunes via supervised loss. A complete trajectory
\[
\tau=\{Q_p,(q_1,r_1),\dots,(q_T,r_T),A'\}
\]
receives
\[
R(\tau)=\alpha\cdot \mathbf 1(A'=A^*)+\beta\sum_{t=1}^T \mathrm{SIG}(q_t).
\]
Stage 2, SIG-Augmented Policy Optimization, builds on Group Relative Policy Optimization and decomposes trajectory reward into action-level signals. Question \(q_t\) receives
\[
R(q_t)=\beta\,\mathrm{SIG}(q_t)+\lambda_q\,w_t\,\mathbf 1(A'=A^*),
\qquad
w_t=\frac{\mathrm{SIG}(q_t)}{\sum_{j=1}^T \mathrm{SIG}(q_j)},
\]
while the final answer receives
\[
R(A')=\lambda_a\,\mathbf 1(A'=A^*),
\qquad
\lambda_q+\lambda_a=\alpha.
\]

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 \(f_{\mathrm{pitting}}\) newly appears in \(U_t\), and if \(\tilde\phi_{\mathrm{pitting}}=0.30\) and no other facts are gained, then \(\mathrm{SIG}(q_t)=0.30\). Experimentally, across three LLMs and two benchmarks, ProMed(Stage 1+2) achieves up to \(55.60\%\) vs. second-best \(51.98\%\) on MedQA and \(59.33\%\) vs. \(51.78\%\) on CMB, with average relative improvement over second-best \(6.29\%\) and a \(54.45\%\) gain over the direct baseline. Ablations that remove SIG or its components degrade performance by up to \(5\!-\!10\) 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 [2606.02247]. Let \(P=\{1,\dots,p\}\) be the player set, \(\nu(S)\) the expensive value of coalition \(S\), and \(\phi=(\phi_1,\dots,\phi_p)^\top\) the vector of Shapley values for \(\nu\). At iteration \(t\), with data
\[
D_t=\{(z^{(1)},y_1),\dots,(z^{(t-1)},y_{t-1})\},
\]
where each \(z^{(i)}\in\{0,1\}^p\) encodes a coalition and \(y_i=\nu(z^{(i)})+\varepsilon_i\), \(\varepsilon_i\sim N(0,\sigma^2)\), the next coalition is chosen to maximize
\[
EIG(z):=I(\phi;y_z\mid D_t)
=
H[\phi\mid D_t]
-
E_{y_z\mid D_t}\bigl[H[\phi\mid D_t\cup\{(z,y_z)\}]\bigr].
\]
Here the quantity is the expected reduction in differential entropy about the Shapley vector after observing \(\nu(z)\).

A Gaussian-process prior is placed on \(f(z)=\nu(z)\) over \(\{0,1\}^p\), with zero mean and a Hamming-distance kernel. Writing \(\theta=f(Z)\in\mathbb R^{2^p}\) for all coalitions, the posterior is
\[
\theta\mid D_t\sim N(\mu_t,\Sigma_t).
\]
By linearity of Shapley values,
\[
\phi=A\theta,
\]
where \(A\in\mathbb R^{p\times 2^p}\) is the fixed matrix of Shapley weights. Hence
\[
\phi\mid D_t\sim N(A\mu_t,\,A\Sigma_tA^\top).
\]
Under this linear-Gaussian structure, mutual information is available in closed form. For a one-point design \(z\) with observation \(y_z=e_z^\top\theta+\varepsilon\), SIG simplifies to
\[
SIG(z)\propto \log[(Me_z)_z]-\log[((M-Q)e_z)_z],
\]
where \(M=\Sigma_t+\sigma^2 I\) and \(Q=\Sigma_tA^\top(A\Sigma_tA^\top)^{-1}A\Sigma_t\). The paper states that \((Me_z)_z\) is the marginal posterior variance at \(z\), and \((M-Q)e_z\) reflects the variance after conditioning on \(y_z\).

Naïve evaluation is exponential in \(p\), because it would require manipulating \(2^p\times 2^p\) matrices. ShaplEIG reduces this to polynomial complexity by exploiting the product structure of the Hamming kernel and the combinatorial structure of \(A\). Elementary symmetric polynomials are used for the linear term \(Ak(Z,z)\), and bivariate generating polynomials with pre- and suffix table convolutions are used for the quadratic term \(AKA^\top\). The resulting complexity is \(O(p^4+t^3)\) for a single candidate and \(O(p^4+t^3+|W|t^2)\) for a batch of \(|W|\) candidates.

The practical loop alternates GP hyperparameter fitting, SIG evaluation over candidate coalitions, and selection of \(z^*=\arg\max_z SIG(z)\). After \(T\) costly evaluations, the posterior mean \(A\mu_T\) yields consistent Shapley-value estimates. Empirically, across nine real-world costly games with \(p=8\ldots 101\) players, ShaplEIG outperforms or matches stochastic, surrogate-based, and fixed-design Bayesian experimental-design baselines, with gains largest in the low-budget regime \(T\ll 2^p\).

## 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 [2306.05724]. In generalized linear models, it is the Shapley decomposition of increments in \(R^2_{\mathrm{KL}}\), a goodness-of-fit quantity normalized to \([0,1]\) [2601.00773]. In ProMed, it is the importance-weighted gain in entailed atomic facts that guides question asking [2508.13514]. In ShaplEIG, it is mutual information between a future coalition observation and the Shapley vector under a GP posterior [2606.02247].

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 \(v\), 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.

Source: https://www.emergentmind.com/topics/shapley-information-gain-sig