S-information: Concepts and Applications

Updated 6 July 2026

S-information is a family of distinct information metrics, ranging from psychophysical mutual information to measures of multivariate interaction and shared redundancy.
In psychophysics, it quantifies the mutual information between stimulus strength and combined responses, leading to robust measures like Shannon competence.
In multivariate analysis, S-information aggregates interactions among variables to disentangle redundancy and synergy, while distinguishing itself from statistical S-values based on P-values.

Searching arXiv for recent and relevant uses of “S-information” to ground the article and disambiguate the term. S-information is not a single universally standardized quantity in the arXiv literature surveyed here. Instead, it names several non-equivalent constructs that share an information-theoretic vocabulary while addressing different inferential problems: psychophysical discriminability from joint reaction-time and choice data, multivariate dependence strength in complex systems, shared or redundant information in partial information decomposition, and the closely related but distinct statistical $S$ -value derived from $P$ -values (Stone, 2021, Bounoua et al., 2024, Bertschinger et al., 2012, Rafi et al., 2020). The term therefore requires domain-specific disambiguation before any mathematical or empirical interpretation is possible.

1. Terminological scope

In the literature considered here, “S-information” appears in at least three explicit senses. In psychophysics it denotes the mutual information between stimulus strength and a joint behavioral response composed of reaction time and binary choice, from which “Shannon competence” $s'$ is derived (Stone, 2021). In multivariate information theory it denotes

$S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$

the average mutual information between each variable and the rest of the system (Bounoua et al., 2024). In partial information decomposition, the letter $S$ is often attached to shared information about a target $S$ , as in $SI(S:X_1;X_2)$ , although those works are primarily about redundancy, uniqueness, and synergy rather than a single canonical scalar called “S-information” (Bertschinger et al., 2012, Rauh et al., 2017).

Usage	Core definition	Representative source
Psychophysical S-information	$I(x;y)$ for joint RT-plus-choice responses	(Stone, 2021)
Multivariate S-information	$\sum_i I(X_i;X_{\setminus i})$	(Bounoua et al., 2024)
Shared-information usage	Redundant information about target $S$	(Bertschinger et al., 2012)

A recurrent source of confusion is that the statistical $P$ 0-value,

$P$ 1

is also an information-like quantity, but it is an information re-expression of a valid $P$ 2-value rather than a mutual-information functional on random variables (Rafi et al., 2020). A second source of confusion is that system-level frameworks such as “System Information Decomposition” and semantic-information formalisms use the letter $P$ 3 for “system” or “semantic,” not for the same quantity (Lyu et al., 2023, Majumdar et al., 2016).

2. Psychophysical S-information and Shannon competence

In "Using Information Theory to Measure Psychophysical Performance" (Stone, 2021), S-information is the mutual information between stimulus strength $P$ 4 and a combined response vector built from reaction time and binary response probability. The paper writes

$P$ 5

with the combined mean response

$P$ 6

where $P$ 7 is the proportion of “comparison” responses at stimulus level $P$ 8, and $P$ 9 is the mean reaction time at that stimulus strength. Under Gaussian assumptions, the entropy of the unconditional response is expressed through the determinant of the covariance matrix $s'$ 0, and the residual entropy after model fitting through the determinant of $s'$ 1, yielding

$s'$ 2

The conditional model used to estimate $s'$ 3 is a covariant extended proportional rate diffusion model (CEPRD), with model response

$s'$ 4

Its mean decision time is

$s'$ 5

total reaction time is $s'$ 6, and the psychometric function is

$s'$ 7

The likelihood is maximized as

$s'$ 8

with the Gaussian likelihood based on the full residual covariance $s'$ 9. The paper presents this as an advance over earlier work that minimized only the product of the RT and choice variances and thereby effectively assumed zero covariance (Stone, 2021).

The discriminability index derived from this S-information is “Shannon competence” $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$0, defined analogously to the relation between $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$1 and information: $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$2 The paper reports an average single-trial information of approximately $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$3 bits, whereas the separate contributions were $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$4 bits for reaction time and $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$5 bits for binary response. Their sum, $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$6 bits, is substantially smaller than the observed joint value, which the paper interprets as synergy between RT and choice. With $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$7 bits, the reported $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$8 is $S(X)\defeq \sum_{i=1}^{N} I(X_i;X_{\setminus i}),$9, compared with $S$ 0 inferred from choice accuracy alone; the corresponding information rate is about $S$ 1 bits/s (Stone, 2021).

The conceptual significance of this formulation is specific. Because $S$ 2 is computed from the joint response $S$ 3, the paper states that it is immune to speed–accuracy trade-offs that can distort $S$ 4-based analyses. It also concludes that using both RT and binary responses can reduce the number of stimulus presentations required while preserving or improving precision in psychophysical parameter estimation (Stone, 2021).

3. Multivariate S-information in high-order dependence analysis

In "S $S$ 5I: Score-based O-INFORMATION Estimation" (Bounoua et al., 2024), S-information is a multivariate interaction measure rather than a psychophysical discriminability statistic. It is defined as

$S$ 6

where $S$ 7. The paper rewrites it as

$S$ 8

where $S$ 9 is total correlation and $S$ 0 is dual total correlation: $S$ 1

Under this definition, S-information quantifies overall interaction strength in a multivariate system, but not whether the dependence structure is redundancy- or synergy-dominated. That balance is instead measured by O-information,

$S$ 2

Positive $S$ 3 indicates redundancy-dominated structure; negative $S$ 4 indicates synergy-dominated structure. For $S$ 5, the paper notes that O-information reduces to co-information or interaction information (Bounoua et al., 2024).

The methodological contribution of the paper is an estimator that removes the usual discrete-variable or Gaussianity restrictions. It uses a diffusion-style noising process,

$S$ 6

and score-based KL-divergence estimation built from time-dependent score functions $S$ 7. The paper states that computing O-information requires $S$ 8 denoising score functions, but amortizes them into a single neural network by feeding the network the noised variables together with a mask or time-vector that encodes which variables are noised and which are clean. Training uses denoising score matching, and inference estimates the time integrals by Monte Carlo with about 10 samples (Bounoua et al., 2024).

The empirical role of S-information in that framework is mainly structural. It is the interaction-strength quantity from which redundancy-like and synergy-like contributions are separated via $S$ 9 and $SI(S:X_1;X_2)$ 0. The paper validates the estimator on synthetic Gaussian and non-Gaussian benchmarks and applies it to mouse visual cortex activity from the Allen Brain Observatory / Visual Behavior Neuropixels dataset. There O-information is generally higher for change than for no-change trials, which the paper interprets as stronger redundant information sharing across visual cortical regions (Bounoua et al., 2024).

The statistical $SI(S:X_1;X_2)$ 1-value discussed in "Technical Issues in the Interpretation of S-values and Their Relation to Other Information Measures" (Rafi et al., 2020) is not the same object as either psychophysical or multivariate S-information. It is defined by

$SI(S:X_1;X_2)$ 2

with $SI(S:X_1;X_2)$ 3 a valid $SI(S:X_1;X_2)$ 4-value. The paper interprets this as surprisal or refutational information against the tested model. A $SI(S:X_1;X_2)$ 5-value of $SI(S:X_1;X_2)$ 6 corresponds to approximately $SI(S:X_1;X_2)$ 7 bits. Alternative logarithmic bases give the same concept in different units: $SI(S:X_1;X_2)$ 8 in nats and $SI(S:X_1;X_2)$ 9 in hartleys, bans, or dits (Rafi et al., 2020).

The crucial technical condition is validity. The interpretation of $I(x;y)$ 0 as information against a model is justified only when the $I(x;y)$ 1-value is uniform under the null or conservatively valid. Under exact uniformity, the $I(x;y)$ 2-value in nats is exponentially distributed with mean 1 nat; in bits, the mean is $I(x;y)$ 3 bits. The paper explicitly rejects posterior predictive $I(x;y)$ 4-values as inputs to this transformation, because they are pulled toward $I(x;y)$ 5 and are not uniform under the null (Rafi et al., 2020).

This usage is adjacent to, but distinct from, mutual-information notions of S-information. The $I(x;y)$ 6-value is a frequentist information score attached to a tail probability; it is not a function of a joint distribution over multiple random variables in the Shannon sense. The paper further distinguishes it from maximum-likelihood-ratio deviance, Bayes factors, Grünwald’s broader safe $I(x;y)$ 7-tests, and severity measures. It also cautions that independent $I(x;y)$ 8-values do not simply add as a direct measure of combined evidence; the correct combined $I(x;y)$ 9-value must come from the $\sum_i I(X_i;X_{\setminus i})$ 0-value of an appropriate summary test statistic (Rafi et al., 2020).

5. Shared information, redundancy, and extractability

A third major usage attaches the letter $\sum_i I(X_i;X_{\setminus i})$ 1 to shared information about a target random variable $\sum_i I(X_i;X_{\setminus i})$ 2. In "Shared Information -- New Insights and Problems in Decomposing Information in Complex Systems" (Bertschinger et al., 2012), the central problem is how to decompose

$\sum_i I(X_i;X_{\setminus i})$ 3

into shared or redundant, unique, and synergistic components. For two sources the intended decomposition is

$\sum_i I(X_i;X_{\setminus i})$ 4

Williams–Beer redundancy is written $\sum_i I(X_i;X_{\setminus i})$ 5, with Möbius inversion used to define local partial terms on the PI lattice (Bertschinger et al., 2012).

That paper is primarily axiomatic. It lists global positivity, weak symmetry, self-redundancy, and monotonicity as Williams–Beer properties, then studies stronger conditions such as strong symmetry and left monotonicity. Its central negative result is that there is no shared-information measure satisfying simultaneously strong symmetry, monotonicity, self-redundancy, and local positivity. The proof uses an XOR construction. The paper also argues that $\sum_i I(X_i;X_{\setminus i})$ 6 can assign large shared-information values in cases where variables are similarly informative without necessarily sharing the same information, and that intuition about independent variables “not sharing information” can fail in game-theoretic examples of shared knowledge (Bertschinger et al., 2012).

"On extractable shared information" (Rauh et al., 2017) modifies this landscape by defining an extractable, left-monotonic closure of a shared-information measure: $\sum_i I(X_i;X_{\setminus i})$ 7 where the supremum is over deterministic functions of the target. The construction guarantees left monotonicity with respect to local processing of the target. It also induces a nonnegative bivariate decomposition by defining the corresponding unique and complementary terms from $\sum_i I(X_i;X_{\setminus i})$ 8 and the mutual informations $\sum_i I(X_i;X_{\setminus i})$ 9, $S$ 0, and $S$ 1. The paper’s main structural result is that left monotonic shared information is not compatible with a Blackwell interpretation of unique information: there is no bivariate decomposition in which shared information is left monotonic and unique information simultaneously satisfies the Blackwell property (Rauh et al., 2017).

These works use “S” in a target-centric way: $S$ 2 is the variable about which information is being shared. They are therefore related to, but mathematically distinct from, the psychophysical and multivariate senses of S-information.

Several additional arXiv works use nearby terminology that can be conflated with S-information but address different objects. "System Information Decomposition" introduces a target-free framework that decomposes a multivariate system’s entropy into external, unique, redundant, and synergistic information atoms, with the central claim that all atoms are symmetric with respect to target choice (Lyu et al., 2023). For three variables, the paper writes

$S$ 3

This is a system-wide decomposition framework rather than a definition of S-information as such (Lyu et al., 2023).

"Semantic Information Encoding in One Dimensional Time Domain Signals" uses semantic information in a geometric sense. There the key operator is

$S$ 4

interpreted as the rate at which kinetic energy is dissipated to encode semantic information in signal shape, and the total semantic information over an interval is

$S$ 5

The paper claims that a digital signal has exactly 13 local geometric configurations in the smallest three-point neighborhood, and that an analog signal has 17 exhaustive local semantic encodings (Majumdar et al., 2016).

A further neighboring concept is stochastic information flow. "Quantifying information flow along a stochastic trajectory" defines a trajectory-level directed quantity $S$ 6 from the time derivative of pointwise mutual information and, for Langevin systems, gives

$S$ 7

This is a trajectory-resolved information-flow increment, not an S-information in any of the preceding senses (Oh et al., 13 May 2026).

The broader implication is terminological rather than doctrinal. Within contemporary arXiv usage, “S-information” is best treated as a family resemblance term whose formal meaning is fixed by context: stimulus-encoding mutual information in psychophysics, system interaction strength in multivariate analysis, or shared-information structure in redundancy decompositions. Statements about its value, invariances, or operational role are therefore not portable across these literatures without first specifying which definition is in use (Stone, 2021, Bounoua et al., 2024, Bertschinger et al., 2012).