Non-Equiprobable Signaling: Theory & Applications

• Non-equiprobable signaling is the process where messages or signals are transmitted with non-uniform probabilities, influencing equilibrium, inference, and performance across various domains.
• In communications, probabilistic shaping assigns tailored probabilities to constellation points to improve error rates and power efficiency, as seen in OFDM and PAM systems.
• In game theory and quantum frameworks, non-equiprobable strategies refine equilibrium selection and contextuality measures by leveraging differential signal frequencies and learning dynamics.

Non-equiprobable signaling refers to processes in which messages, symbols, or measurement settings are selected or transmitted with non-uniform probability distributions. The concept appears prominently across economics (signaling games), classical and quantum probability theory, and modern digital communications. Non-equiprobable signaling modifies system behavior and information properties in ways that equiprobable (uniform) signaling cannot, with direct implications for equilibrium selection, inference, statistical discrimination, and capacity optimization.

1. Non-Equiprobable Signaling in Game Theory and Learning

In dynamic signaling games, non-equiprobable signaling endogenously arises from Bayesian learning dynamics among agents with heterogeneous types and payoff structures. Specifically, in the micro-foundation developed by Fudenberg and He, each sender type θ selects signals s∈S according to individualized optimal experimentation rules shaped by the Gittins index:

$$G_{θ}(s) = \sup_{\tau>0} \frac{\mathbb{E}\left[\sum_{t=0}^{\tau-1}\beta^t u_1(\theta, s, a_s(t))\right]}{\mathbb{E}\left[\sum_{t=0}^{\tau-1}\beta^t\right]}$$

where β is an effective discount factor, and u₁ is the sender's payoff. Payoff differentials induce persistent differences ("Gittins gaps") in these indices across types, and thus, in the steady state, the frequency $p_{θ}(s)$ with which each type sends each signal is non-uniform, even off the equilibrium path. This directly constrains receivers’ possible beliefs about off-path behavior:

$$p_{i}(s) \geq p_{j}(s) \Rightarrow \mu(s)[i] \geq \mu(s)[j]$$

where μ(s)[i] is the posterior belief assigned to type i after receiving signal s. These constraints yield strict equilibrium refinements, eliminating pooling equilibria that rely on counterfactual "perverse" off-path beliefs, and ensuring that observed frequencies of signal use reflect the true "type compatibility" of senders and signals (Fudenberg et al., 2017).

2. Non-Equiprobable Signaling in Wireless Communications: Shaping and Capacity

In digital communications, non-equiprobable signaling—typically called probabilistic shaping—is used to assign unequal probabilities to constellation points or subconstellations, exploiting channel properties for increased spectral efficiency or reduced power requirements.

For orthogonal frequency-division multiplexing (OFDM) systems, constellations (e.g., M-QAM) are partitioned into K annular subconstellations ("rings"), each with L points. A shaping code selects rings with tailored frequencies π₁,…,π_K, assigning lower probability to high-energy rings. Integration with LDPC coding involves:

• The LDPC code determines a base symbol (e.g., gray-labeled QAM point in the inner ring).
• The shaping code selects which ring’s representative to transmit, according to the desired π distribution.

Calculation of log-likelihood ratios (LLRs) in the LDPC decoder explicitly incorporates the prior probabilities of constellation points set by the shaping code:

$$L(b|y) = \log \frac{\sum_{s: \ell_b(s)=0} p(y|s)\pi(s)}{\sum_{s: \ell_b(s)=1} p(y|s)\pi(s)}$$

Simulations show that shaping at 1.25–1.5 bits/symbol in 4-QAM provides 4–5 dB BER gain at $10^{-3}$, and for 16-QAM, shaping yields 1 dB gain at equivalent error rates. The performance advantage diminishes as rate approaches the entropy of the constellation (Mattu et al., 6 Dec 2025).

A related approach, using truncated-Gaussian-distributed pulse-amplitude modulation (PAM) points and merging close points to form "non-equiprobable and non-uniform (NENU)" constellations, achieves optimal trade-offs between mutual information, peak-to-average power ratio (PAPR), and decoding complexity. Constellation parameters (truncation parameter ρ, cardinality N, and clipping ratio γ_CR) are optimized to maximize bit-wise mutual information (BMI) under PAPR and SNR constraints, and deliver substantial BER and PAPR improvements over both uniform and DFT-precoded single-carrier designs (Kurihara et al., 21 Aug 2025).

Approach	Key Mechanism	Quantitative Gains
Ring Shaping + LDPC	Binary shaping code for rings; priors in LLR	4–5 dB (4-QAM), 1 dB (16-QAM), BER at $10^{-3}$ (Mattu et al., 6 Dec 2025)
Truncated-Gaussian PAM	Shaping+merging, CAF+CNC	BMI increase by 0.2 bits/dim, PAPR reduction by 5 dB, 2 dB BER gain (AWGN/fading) (Kurihara et al., 21 Aug 2025)

3. Non-Equiprobable Signaling in (No-)Signaling Quantum and Classical Models

Marginal selectivity violation (signaling) occurs when the marginal probability of one observable depends on other measurement settings:

$⟨A_{i1}⟩ \neq ⟨A_{i2}⟩$

Such non-equiprobable signaling—where violations can vary in magnitude—breaks standard Bell or Leggett-Garg inequalities, which assume marginal invariance. Dzhafarov and Kujala generalize these inequalities by quantifying the minimal connection-deviation C₀ permitted by the observed marginal differences:

$$C_0 = \left(\frac{1}{2}|⟨A_{11}⟩-⟨A_{12}⟩|, \ldots \right)$$

and replacing the CHSH bound by $2(1 + \Delta_0)$, with $\Delta_0 = \sum C_0$. The degree of contextuality is then defined as the excess joint-incompatibility unexplained by signaling. This measure is continuous in correlators and robust for finite data, restoring a sharp boundary between contextually and non-contextually explainable correlations in the presence of arbitrary—potentially non-uniform—signaling (Dzhafarov et al., 2014).

Khrennikov and Alodjants construct a classical Kolmogorovian probability model where selection probabilities $p_{ij}$ for measurement settings are allowed to be non-uniform. Conditional (epistemic) correlations remain well-defined and independent of $p_{ij}$, while unconditional (ontic) correlations and CHSH-type sums are weighted by $p_{ij}$. The correct identification of empirical data with conditional rather than unconditional correlations ensures that no-signaling properties are determined by marginal independence rather than equiprobability of settings (Khrennikov et al., 2018).

4. Theoretical Foundations and Frameworks: Kolmogorov vs Frequentist Approaches

Standard quantum theory, under Kolmogorov's measure-theoretic probability, guarantees ensemble indistinguishability: any two decompositions of the same density matrix yield identical observable statistics, preserving the no-signaling principle. In contrast, frequentist-inspired frameworks (à la von Mises) admit content-dependent fluctuations (denoted κ) in empirical frequencies, potentially breaking this indistinguishability. Physical protocols exploiting such intrinsic fluctuation differences—when ensembles realize the same density but with different preparation histories—permit discrimination and even superluminal communication, violating the no-signaling theorem (Kumar et al., 2019).

Probability Framework	No-Signaling Guarantee	Distinguishability of Ensembles
Kolmogorov (KQM)	Indistinguishability of all decompositions	No; ensembles are operationally identical (Kumar et al., 2019)
Frequentist (FQM)	Fluctuations in limiting frequencies; possible signaling	Yes; distinguishable via fluctuation signatures (Kumar et al., 2019)

5. Implications, Trade-offs, and Applications

Non-equiprobable signaling is leveraged for both performance enhancement and theoretical refinement. In communications, shaping provides near-capacity gains and PAPR control without hardware changes, at the price of increased demapping complexity and stringent codebook management. In signaling games, learning-driven non-equiprobable experimentation resolves equilibrium indeterminacy and constrains off-path inference.

In foundational physics, non-equiprobable signaling challenges standard constraints, both as an artifact of experimental imperfections (requiring generalized contextuality measures) and as a (hypothetical) consequence of adopting frameworks with intrinsic frequency fluctuations.

Designing for or against non-equiprobable signaling is inherently context-dependent:

• In wireless channels, shaping codes are tuned to system requirements (rate, power, complexity).
• In quantum contextuality and signaling analyses, non-equiprobable selection probabilities and marginal-driven deviations drive the choice of theoretical tools (conditional probability, joint coupling, and generalized inequalities).
• In economic signaling, the frequency of out-of-equilibrium signals informs equilibrium refinement via type-compatibility, altering the strategic landscape.

6. Illustrative Examples and Quantitative Results

• In learning-based signaling games (Fudenberg et al., 2017), sender types with strictly higher Gittins indices for a given signal s will, in steady state, send s more frequently, imposing posterior inequalities on receiver beliefs and refining Nash equilibria.
• In 4-QAM OFDM systems, non-equiprobable ring shaping at 1.5 bits/sym achieves 4 dB BER improvement at $10^{-3}$ over uniform signaling, with smaller but non-negligible gains for larger constellations (Mattu et al., 6 Dec 2025).
• Bell-system experiments admit explicit calculation of contextuality degrees under arbitrary marginal deviations (non-equiprobability of marginals), preserving noncontextuality up to the point where the generalized CHSH bound is violated (Dzhafarov et al., 2014).
• In frequentist quantum frameworks, the possibility of signaling—previously ruled out solely by the indistinguishability of ensemble decompositions—arises via content-dependent limitations of empirical frequencies, even when ensemble densities are identical (Kumar et al., 2019).

7. Open Problems and Research Directions

While non-equiprobable signaling is practically exploited and theoretically quantified in many contexts, several issues remain open:

• Extension of generalized contextuality measures to multi-setting and higher-dimensional observables (Dzhafarov et al., 2014).
• Statistical inference of non-equiprobable signaling under experimental noise and finite data.
• Exploration of block-length, code weight, and ring-number trade-offs for optimal shaping in adaptive communications (Mattu et al., 6 Dec 2025).
• Physical implementation and possible empirical tests distinguishing Kolmogorov and frequentist quantum frameworks (Kumar et al., 2019).
• Further unification of economic and physical models for inference and learning under non-equiprobable signaling regimes.

Non-equiprobable signaling thus constitutes a key structural feature in the modeling, optimization, and interpretation of modern learning, inference, and communications systems.