Belief Entrenchment: Bayesian & Social Dynamics

• Belief entrenchment is a tendency of agents to reinforce initial views despite contradictory evidence, undermining rational updates.
• Mathematical models like the Bayesian Martingale Score capture how prior beliefs predict overconfident updates in both individuals and networks.
• Empirical and simulation studies show that targeted mitigation strategies, including independent evidence sharing and open-minded protocols, can reduce excessive entrenchment.

Belief entrenchment refers to a systematic and quantifiable propensity of agents—human, artificial, or collective—to reinforce prior convictions instead of updating rationally in light of new, potentially conflicting evidence. Across cognitive science, social simulation, Bayesian learning, and knowledge representation, belief entrenchment is implicated in phenomena ranging from individual confirmation bias to persistent polarization in social systems and pathological failure modes in LLMs. Its formalization extends from Bayesian statistics, active inference, social network theory, to paraconsistent logics. This article surveys the mathematical, computational, and empirical foundations of belief entrenchment, identifies its core mechanisms, and discusses approaches for detection and mitigation.

1. Formal Definitions and Bayesian Characterization

Belief entrenchment is defined in "Martingale Score: An Unsupervised Metric for Bayesian Rationality in LLM Reasoning" (He et al., 2 Dec 2025) as a violation of the Bayesian Martingale property during iterative reasoning. For any stochastic process $(b_t)$ denoting agent beliefs at discrete steps, perfect Bayesian updating assures that expected future belief equals the current belief, i.e., $\mathbb{E}[b_{t+1}|b_t]=b_t$. Under this property, belief updates $\Delta b = b_{t+1}-b_t$ are not systematically predictable from $b_t$ alone. Entrenchment is present when high (low) prior beliefs $b_\mathrm{prior}$ systematically lead to further upward (downward) belief updates $\Delta b$—i.e., the regression slope between $\Delta b$ and $b_\mathrm{prior}$ (Martingale Score $M$) is significantly positive.

Entrenchment undermines rational inference: instead of letting newly arrived data re-calibrate or overturn prior convictions, agents reinforce their initial guess, generating overconfident but poorly justified inferences. This principle generalizes to active inference (Catal et al., 2 Jul 2024), where belief entrenchment occurs if variational posteriors $q_t(s)$ become increasingly narrow in the absence of new external evidence, as quantified by entropy reduction $E_t=H[q_{t-1}]-H[q_t]\geq 0$.

2. Mechanisms and Mathematical Models

Entrenchment is emergent in diverse dynamical systems. In LLMs, multi-step reasoning (e.g., Chain-of-Thought, Debate) systematically relates the trajectory of belief updates to their initial values, well captured by the Martingale Score $M$—a consistent estimator for entrenchment across domains and prompt regimes (He et al., 2 Dec 2025). In multi-agent setups, explicit belief strengths ($v_i$) and open-mindedness coefficients ($\lambda$) formalize entrenchment and resistance to change (Bilgin et al., 6 Dec 2025).

For belief networks embedded in social graphs, entrenchment is modeled by competing cognitive and social coherence drives. Internal consistency is penalized via an “energy” function over networks of signed beliefs $a_{jk}\in\{\pm1\}$, while social conformity is encoded by alignment with neighbors (Rodriguez et al., 2015). Balance is expressed through Hamiltonians $H = \sum_n [J E^{(i)}_n + I E^{(s)}_n]$, with $J$ (coherentism) and $I$ (peer influence) as parameters.

Gradient-based update rules under belief certainty trade-offs ($w_i$ as an inverse-variance function of internal/external coherence) push belief systems toward extreme entrenchment: ideological alignment (internal coherence) or social alignment (network coherence), with positive feedback loops preventing intermediate states (Hewson et al., 7 Oct 2024).

3. Empirical Detection and Quantification

Reasoning systems, especially LLMs, exhibit pervasive beliefs entrenchment as demonstrated by regression-based metrics. The Martingale Score $M$ is calculated by fitting $$\Delta b_i = \beta_0 + \beta_1 b_{\mathrm{prior},i} + \epsilon_i$$, with $M=\hat\beta_1$, where $M>0$ signals reinforcement of priors ($b_\mathrm{prior}$ predicts update $\Delta b$) (He et al., 2 Dec 2025).

Domain-agnostic experiments show that almost all tested Chain-of-Thought LLM runs (51/54) yield $M>0$, with ground-truth calibration (Brier scores) deteriorating as $|M|$ increases. Prompt engineering only modestly mitigates entrenchment; instructions to prioritize prior-consistent arguments further exacerbate it.

In multi-agent LLM debates using belief-box formalism, degree of entrenchment $E_\alpha(b_i)\equiv v_i^\alpha$ is empirically manipulated through the open-mindedness parameter $\lambda_\alpha$ (Bilgin et al., 6 Dec 2025). Entitlement to change is measured by belief-change rates and persuasiveness across agent groups. Strong initial epistemic strengths confer marked resistance against peer pressure; only overwhelming group consensus can break entrenchment.

Active inference agents, when sharing raw posteriors, inevitably converge to degenerate distributions (echo chambers), amplifying tiny initial biases (Catal et al., 2 Jul 2024). Likelihood-only sharing mitigates overconfidence, restoring sensitivity to environmental evidence.

4. Social and Network Dynamics of Entrenchment

Population-level phenomena emerge from the interplay between cognitive coherence and social conformity (Rodriguez et al., 2015). Strong internal coherence ($J\gg I$) generates rigid, jammed clusters; strong social pressure ($I\gg J$) drives fast consensus. Minority “zealots” with perfectly coherent belief networks resist majority invasion, producing stable, entrenched belief clusters even under intense exposure.

Simulation-based studies confirm that belief dynamics driven by dissonance reduction settle in one of two entrenched extremes: full internal or full social alignment (Hewson et al., 7 Oct 2024). The absence of negative feedback mechanisms ensures that intermediate, less polarized states are unstable. Modifications to the update rule, such as open-mindedness floors or adaptive social ties, can reintroduce ongoing dynamics rather than maximal entrenchment.

5. Entrenchment in Logics and Belief Revision

Epistemic entrenchment is central to rational belief revision. In the paraconsistent logic RCBr (Coniglio et al., 9 Dec 2024), entrenchment is encoded as a binary relation $\le_K$ on a belief set $K$, satisfying transitivity, dominance, conjunctiveness, minimality, and maximality. Strongly accepted beliefs ($o a\in K$) are maximally entrenched and unrevocable. Contraction and revision operators $C_K(K,a)$ and $K*_K a$ are defined in terms of $\le_K$, preserving more entrenched beliefs during contradiction or revision steps. Representation theorems ensure correspondence between contraction postulates and entrenchment relations.

Non-deterministic matrix semantics (Nmatrices) and Boolean algebras with LFI operators (BALFIs) provide algebraic and semantic foundations for ranking belief entrenchment and facilitating rational contraction in inconsistent knowledge bases.

6. Mitigation Strategies and Future Research Directions

Mitigation of belief entrenchment is multifaceted. For LLMs, critical reflection prompts only modestly reduce entrenchment; more effective are design protocols that prioritize true evidence integration over prior reinforcement (He et al., 2 Dec 2025). In agent-based simulations, increased open-mindedness ($\lambda$) or argument force ($a$) decrease entrenchment; conversely, high $\lambda$ and strong peer pressure can induce belief change.

Communication protocols in multi-agent systems should emphasize sharing independent evidence (likelihoods) rather than re-broadcasting posteriors to avoid echo chambers and runaway overconfidence (Catal et al., 2 Jul 2024). Negative feedback mechanisms—open-mindedness floors, stochastic noise, dynamic ties, or bounded-confidence thresholds—are essential to avert collapse into extreme entrenchment configurations (Hewson et al., 7 Oct 2024).

In paraconsistent belief revision, entrenchment-based formalism allows flexible contraction and revision, ranking beliefs to preserve core information while retracting only minimally entrenched assumptions (Coniglio et al., 9 Dec 2024).

7. Broader Implications

Belief entrenchment is a foundational failure mode in cognitive, social, and artificial reasoning systems. Its mathematical formalization provides tools for quantification, diagnosis, and mitigation. Entrenchment explains persistent polarization, the resistance of coherent minorities (“zealots,” cults), and unreliable inference in LLMs. Models integrating cognitive and social forces (Rodriguez et al., 2015, Hewson et al., 7 Oct 2024), Bayesian rationality metrics (He et al., 2 Dec 2025), and logical contraction operators (Coniglio et al., 9 Dec 2024) collectively advance understanding and effective design of truth-seeking systems.

A plausible implication is that future models and deployed reasoning agents should integrate adaptive feedback, dynamic openness, and principled evidence sharing to maintain fidelity to truth rather than ritualized self-affirmation. Failure to address entrenchment risks undermining the reliability, adaptability, and social utility of both human and artificial cognitive architectures.