Public Belief States (PBS)

• Public Belief States (PBS) are distributions over collective Bayesian posteriors that encapsulate individual updates from private signals and common priors.
• PBS feasibility is characterized by a mean-preserving spread condition, providing both necessary and sufficient criteria under specified information structures.
• Dynamic PBS models extend to capture divergences between private and public beliefs, informing analyses on polarization, persuasion, and false public realities.

Public Belief States (PBS) represent distributions over the collective pattern of posterior beliefs in a population of agents, typically where individuals update their beliefs based on private information and a shared common prior. PBS concepts codify both the range of possible aggregate belief configurations and the constraints imposed by information structures and updating protocols, underpinning rigorous analyses of polarization, consensus, and social inference phenomena.

1. Formal Definition and Mathematical Structure

Consider a finite state space $\Omega = \{1, \dots, m\}$ with a common prior $\mu \in \Delta(\Omega)$ across a population of $n$ Bayesian agents. Each agent $i$ receives a private signal $s_i$ and computes a posterior $x_i(s_i) \in \Delta(\Omega)$. For a full signal profile $s = (s_1, \dots, s_n)$, the empirical distribution of posteriors is defined as:

$$H_s = \frac{1}{n} \sum_{i=1}^n \delta_{x_i(s_i)} \in \Delta_n(\Delta(\Omega)),$$

where $\Delta_n(\Delta(\Omega))$ is the set of empirical measures with $n$ atoms on $\mu \in \Delta(\Omega)$0. An information structure $\mu \in \Delta(\Omega)$1 induces a distribution $\mu \in \Delta(\Omega)$2 over such empirical distributions. A measure $\mu \in \Delta(\Omega)$3 is called a feasible Public Belief State if there exists some information structure $\mu \in \Delta(\Omega)$4 yielding $\mu \in \Delta(\Omega)$5 under prior $\mu \in \Delta(\Omega)$6 and population size $\mu \in \Delta(\Omega)$7 (Arieli et al., 2022).

2. Characterization: The Feasibility Theorem

A central result is a necessary and sufficient condition for PBS feasibility, formalized via the concept of a mean-preserving spread (m.p.s.):

• For any $\mu \in \Delta(\Omega)$8, define the ex-ante average $\mu \in \Delta(\Omega)$9 by integrating $n$0 against $n$1.
• For each $n$2, define the conditional-on-state average $n$3 as:

$n$4

• The baseline distribution is $n$5.

Theorem (Arieli & Babichenko):

A distribution $n$6 is a feasible PBS if and only if $n$7 is a mean-preserving spread of $n$8; that is, there exist measures $n$9 on $i$0 such that each $i$1 has barycenter $i$2 and $i$3. This condition is both necessary (arising from partitioning posteriors by state and applying Bayes’ rule) and sufficient (by constructing $i$4 via sampling from $i$5 conditioned on state) (Arieli et al., 2022).

3. Empirical Models and Dynamic PBS

In kinetic-exchange models, such as in Roy & Biswas (2021), PBS are operationalized via coupled public and private opinion variables per agent: a private belief $i$6 and public opinion $i$7. Their evolution is governed by stochastic update equations which incorporate self-conviction, peer influence, and noise: \begin{align*} \sigma_i^{int}(t+1) &= \text{sign}\left[\mu\,\sigma_i^{int}(t) + (1{-}\mu)\sum_j J_{ij} \sigma_j^{int}(t) + \eta_i^{int}(t)\right] \ \sigma_i^{pub}(t+1) &= \text{sign}\left[\nu\,\sigma_i^{pub}(t) + (1{-}\nu)\sum_j K_{ij} \sigma_j^{pub}(t) + \lambda\,\sigma_i^{int}(t) + \eta_i^{pub}(t)\right] \end{align*} The population-level public and private PBS are encapsulated by the magnetizations $i$8 and $i$9. The misalignment

$s_i$0

serves as an order parameter for “opinion hypocrisy” or spiral-of-silence effects, capturing the gap between internal and stated beliefs (Roy et al., 2021).

4. Mean-Preserving Spread in PBS Feasibility

The mean-preserving spread (m.p.s.) framework is critical for distinguishing feasible PBS distributions. In the binary state and binary posterior case (e.g., posterior support only at $s_i$1 and $s_i$2), the feasibility reduces to:

• The law $s_i$3 on the fraction $s_i$4 of agents at $s_i$5 is feasible if and only if $s_i$6 is a mean-preserving spread of a corresponding two-point distribution determined by the prior and posterior supports (Arieli et al., 2022).

Quantile conditions further reduce checking m.p.s. feasibility in such cases to tractable computations involving expected values and lower quantiles. Product settings (where marginals are fixed) yield multinomial PBS, with feasibility characterized analogously via multinomial mean-preserving spread conditions.

5. PBS in Belief Dynamics and Social Perception

Belief dynamics models such as the Personal, Expressed, and Social Beliefs (PES) meta-model embed PBS in a larger architecture incorporating both private (personal), public (expressed), and social (perceived) components:

• Each agent $s_i$7 maintains a personal belief $s_i$8, an expressed belief $s_i$9, and for each neighbor $x_i(s_i) \in \Delta(\Omega)$0 a social belief $x_i(s_i) \in \Delta(\Omega)$1 (agent $x_i(s_i) \in \Delta(\Omega)$2’s perception of $x_i(s_i) \in \Delta(\Omega)$3).
• Social beliefs are updated stochastically to minimize a dissonance potential $x_i(s_i) \in \Delta(\Omega)$4, with Glauber-type update rules.
• This formalism unifies classical models (Voter, Ising, DeGroot, Hegselmann-Krause) as limiting cases and generalizes them to regimes with explicit misexpression (e.g., $x_i(s_i) \in \Delta(\Omega)$5) and misperception ($x_i(s_i) \in \Delta(\Omega)$6), enabling the emergence of false public realities (pluralistic ignorance) (Zimmaro et al., 20 Feb 2025).

6. Applications: Polarization, Persuasion, and Social Misalignment

PBS concepts enable precise characterization of phenomena such as:

• Polarization Maximization: The most polarized feasible PBS occurs when information structures produce full revelation for half the agents and none for the rest. For even $x_i(s_i) \in \Delta(\Omega)$7 and prior $x_i(s_i) \in \Delta(\Omega)$8, the maximal polarization metric $x_i(s_i) \in \Delta(\Omega)$9 is $s = (s_1, \dots, s_n)$0 (Arieli et al., 2022).
• Bayesian Persuasion: In homogeneous settings (binary action, agents adopt if posterior $s = (s_1, \dots, s_n)$1), PBS feasibility identifies the profiles maximizing a sender’s surplus, with the optimal value converging to $s = (s_1, \dots, s_n)$2 as $s = (s_1, \dots, s_n)$3 (Arieli et al., 2022).
• Public-Private Discrepancy: Kinetic and meta-models predict strong divergence between aggregate public (voiced) states and the true distribution of private beliefs. For example, in the PES model, the climate policy case yields macroscopic underestimation of supporter prevalence due to social and expressive divergence mechanisms (Zimmaro et al., 20 Feb 2025).
• Critical Behavior: The transition to consensus, fragmentation, or polarized order in PBS is determined by disorder parameters, private-public coupling, and noise, with critical exponents and thresholds characterized analytically (Roy et al., 2021).

7. Empirical and Theoretical Implications

PBS theory provides both positive (what distributions can arise) and normative (maximal or minimal polarization, susceptibility to persuasion) characterizations. Empirically, the misalignment parameter $s = (s_1, \dots, s_n)$4 can in principle be extracted from combinations of public polling and private (e.g., secret ballot) data, revealing the “hidden public” substratum beneath overt consensus. In belief-dynamics contexts, PBS formalism quantifies and predicts the conditions for phenomena such as pluralistic ignorance, shy-voter effects, and polling bias. The mean-preserving spread condition supplies a tractable and generalizable benchmark for feasible cross-sectional belief configurations.

A plausible implication is that rigorous PBS analysis is essential for interpreting empirical opinion distributions, inferring underlying information structures, and designing interventions (e.g., persuasion campaigns) that rely on precise belief manipulation or measurement (Arieli et al., 2022, Zimmaro et al., 20 Feb 2025, Roy et al., 2021).