Belief Inertia: Minimal Updating & Consensus

• Belief inertia is defined as the resistance to change established beliefs, formalized through inertial updating that minimizes deviation from prior probabilities.
• The models incorporate axiomatic and variational methods, explaining phenomena like confirmation bias, polarization, and strategic inertia in multi-agent systems.
• Computational and network frameworks demonstrate how minimal adjustments, cost trade-offs, and social influence drive consensus formation despite conflicting new evidence.

Belief inertia denotes the resistance that an agent, artificial system, or collective exhibits toward changing established beliefs in the face of new evidence or argument. This phenomenon has been formalized across diverse disciplines—including decision theory, cognitive science, logic, social modeling, multi-agent systems, and machine learning—through axiomatic, algorithmic, and statistical-physics frameworks. The following exposition systematically presents the technical definitions, behavioral principles, formal mechanisms, and practical consequences of the Belief Inertia Argument, drawing on findings from logic, economics, social dynamics, and computational learning theory.

1. Formalizations of Belief Inertia: Distance-Minimization and Updating Principles

In modern decision theory, belief inertia has been formalized via the principle of inertial updating: when faced with new information or events, a decision maker (DM) updates her prior belief $\mu \in \Delta(S)$ (where $S = \{s_1, ..., s_n\}$ is a finite state space) to the new belief $\pi$ by minimizing a subjective distance to the prior, constrained to the set of feasible posteriors. Given observed event $E \subset S$, the DM selects

$\pi_E = \arg\min_{\pi \in \Delta(E)} d_\mu(\pi),$

with $d_\mu$ a strictly convex, prior-centric "distance" function. This framework unifies Bayesian, non-Bayesian, and zero-probability event updates, including Myerson's conditional probability systems (CPS) and distorted (biased) updating rules, by varying the choice of $d_\mu$(Dominiak et al., 2023).

For general (set-valued) information $I \subset \Delta(S)$—including interval, qualitative, and non-standard constraints—inertial updating prescribes

$p^* = \arg\min_{p \in I} D(p\Vert\pi),$

where $D$ is a divergence, often from the $f$-divergence family, selected based on behavioral axioms such as compliance, betweenness, and dynamic consistency(Dominiak et al., 2 Feb 2025).

Fundamental behavioral consequences of inertial updating include:

• Minimal adjustment: Agents make the smallest possible shift from prior belief compatible with new information.
• Endogenized biases: Choosing parametric distortions (e.g., Grether's $\alpha$-rule or S-shaped distortions) recovers empirically observed updating pathologies (e.g., confirmation bias, wishful thinking, motivated reasoning).
• Unification: Both classical Bayesian updating ($d_\mu =$ KL divergence) and non-Bayesian variants (distance-minimized subject to more general $I$ or alternative divergences) fall under the inertial umbrella.

2. Cognitive, Variational, and Resource-Rational Accounts

Resource-rational (variational) frameworks model belief inertia as the outcome of an explicit trade-off between the "utility" of a belief state and the cognitive/pragmatic cost (quantified via Kullback-Leibler divergence) of moving away from the prior(Hyland et al., 22 Sep 2025):

$$F[q, o] = \mathcal{U}[q, o] - \lambda D_{KL}(q\Vert p),$$

where $p$ is the prior, $q$ the candidate posterior, and $\mathcal{U}[q, o]$ combines affective/social and evidential (accuracy) utilities. The parameter $\lambda$ controls inertia: large $\lambda$ immobilizes belief updating ("stubbornness"), small $\lambda$ allows immediate adoption of the belief maximizing $\mathcal{U}$. The optimal variational posterior is

$$q^*(s) \propto p(s) \exp\left(\frac{1}{\lambda}\frac{\delta\mathcal{U}}{\delta q(s)}\right).$$

With substantial $\lambda$, agents display:

• Belief inertia: Small belief shifts even under strong evidence.
• Confirmation bias and polarization: Selective search and weighting of evidence; persistent divergence across agents with differing motivational utilities.
• Empirical consequences: Fit to behavioral update curves allows direct measurement of inertia parameters; interventions that reduce costs or increase evidence salience can mitigate inertia.

3. Social and Collective Dynamics: Inertia in Networks and Consensus Formation

Statistical physics and network models operationalize belief inertia as intrinsic resistance embedded at the node (agent) level. In random-field Ising-type models, each agent's individual predisposition $h_i$ biases their binary "spin" (belief state), so collective consensus is possible only if social coupling (peer influence $J_{ij}$) overcomes personal inertia(Galesic et al., 2017):

$$H(\mathbf{s}) = -\sum_{\langle i,j\rangle} J_{ij} s_i s_j - \sum_i h_i s_i.$$

The kinetic update rule (Glauber dynamics) modifies $s_i$ with probability $\propto \exp(-\beta \Delta H_i)$, making belief flips exponentially unlikely against strong $h_i$.

In multi-concept networks, belief evolution is driven by competition between internal coherence ($J$) and social conformity ($I$), resulting in complex phase diagrams with regimes of consensus, disorder, and long-lived metastable coexistence. Zealots with high internal coherence can tip collective belief once their proportion exceeds a critical threshold (e.g., $10\%$ for fully coherent zealots in aggressive social environments)(Rodriguez et al., 2015).

Opinion inertia also appears in threshold/voter-like models with "stickiness"—an agent requires multiple consecutive exposures to opposing views before switching opinions. This leads to tipping point phenomena: minorities with higher stickiness can overturn majorities if exceeding a calculable critical fraction $p_c$, often dramatically smaller than $0.5$ as stickiness increases(Doyle et al., 2014).

4. Logical, Non-Monotonic, and Argumentative Perspectives

In the context of knowledge bases and non-monotonic reasoning, belief inertia manifests as the resistance of an established (defeasible) argument structure to revision. When new arguments are inserted into a defeasible logic program (DeLP), simply expanding the rule base rarely suffices to make the new argument undefeated (warranted). It is necessary to minimally contract the set of defeasible rules, guided by criteria such as set-inclusion minimality, alteration set minimality, and incision-aware minimality, to ensure the new conclusion emerges as warranted. The process is algorithmically challenging due to the need to balance minimal disruption (preserving unrelated conclusions) with effective overcoming of dialectical opposition(Moguillansky et al., 2011).

In modal logic approaches to belief representation, the principle of inertia asserts: "A belief is preserved over time unless there is a belief to the contrary." This forms a key axiom in reasoning tasks—such as the Sally–Anne false-belief test—requiring that beliefs persist by default, unless actively updated or contravened by negative evidence. The principle is formalized in hybrid modal logic as:

$$B\varphi(t) \wedge t < u \wedge \neg B\neg\varphi(u) \to B\varphi(u),$$

where $B$ is a belief modality and $t,u$ are temporal indices(Brauner, 2013).

Failure to encode such inertia (or its inversion) can lead to non-reversibility in ranked preferential models (RPMs) of belief revision: qualitative models cannot capture the full reversibility that numerical (e.g., Spohn function) approaches afford. Without numerical strengths, belief changes are not fully invertible—a key limitation of qualitative RPM-based models(Hunter, 2013).

5. Empirical and Computational Evidence

Empirical studies across psychology and behavioral economics document robust belief inertia effects:

• Once agents "take sides," they display confirmation bias, overconfidence, and identity-protective cognition, leading to extreme and persistent beliefs.
• Laboratory experiments find that categorical belief commitment drastically diminishes analytic competence, inflates effect sizes (e.g., in conjunction and framing effects), and amplifies polarization, even among highly numerate or educated populations(Martins, 2015).

Computationally, belief inertia is now recognized as a source of worst-case regret in non-stationary reinforcement learning. When historical empirical averages (empirical beliefs) aggregate many past interactions, substantial "momentum" develops, and exceptionally large numbers of novel observations are needed to adjust the belief toward new optimality. In multi-armed bandit problems, this inertia can be adversarially exploited to create problem instances where classical algorithms (Explore-Then-Commit, $\epsilon$-greedy, UCB) suffer linear-in-$T$ regret after a change, regardless of parameter tuning(Mendelson et al., 6 Nov 2025). Even periodic restarts cannot eliminate this regret floor if the number of changes outpaces the restart frequency.

Manifestation	Formalization	Key Reference
Inertial Updating	$\pi_E = \arg\min_{\pi \in \Delta(E)} d_\mu(\pi)$	(Dominiak et al., 2023)
Resource-Rational Variational	$$q^*(s) \propto p(s)\exp\big((c_s+\alpha\log p(o|s))/\lambda\big)$$	(Hyland et al., 22 Sep 2025)
Network/Opinion Inertia	$H = -\sum_{ij}J_{ij}s_is_j - \sum_i h_is_i$	(Galesic et al., 2017)
Stickiness (Threshold Models)	Need $w_A$ exposures before switch; $p_c$ tipping threshold	(Doyle et al., 2014)
Logical Inertia (Modal Logic)	$$B\varphi(t) \wedge t<u \wedge \neg B\neg\varphi(u) \implies B\varphi(u)$$	(Brauner, 2013)
Algorithmic Regret	Sample mean inertia induces $O(T)$ regret after changes in rewards	(Mendelson et al., 6 Nov 2025)

6. Consequences for Rationality, Science, and Intervention Strategies

The belief inertia argument has direct implications for scientific practice, group decision-making, and the design of interventions:

• Probabilistic education: Advocates the continuous assignment and Bayesian updating of probabilities rather than categorical commitment, to minimize inertia-derived extremism(Martins, 2015).
• Scientific reliability: Institutionalizes the ranking of theories by likelihood or posterior probabilities and discourages binary rejection-acceptance rituals, which feed inertia and undermine credibility.
• Persuasion and policy: The "curvature" of the distance/energy function underlying inertia critically shapes the effectiveness of persuasion and information campaigns (e.g., in vaccination or political mobilization), as only sufficiently large, coordinated exposure (exceeding a model-determined tipping threshold) can shift mass beliefs(Doyle et al., 2014).
• Algorithmic modification: In online learning and reinforcement learning, inertia-aware strategies (e.g., explicit forgetting, weighted averaging with decay, adaptive restarts) are necessary to avoid adversarial exploitation(Mendelson et al., 6 Nov 2025).
• Interventions: Empirically, measured inertia parameters can guide individualized interventions—gradual, multi-step exposure and social support reduce effective update costs and overcome otherwise persistent belief inertia(Hyland et al., 22 Sep 2025).

7. Theoretical Unification and Open Problems

Belief inertia arises generically when belief revision is governed by cost minimization, coherence-seeking, or momentum principles—whether cognitive, social, or algorithmic in origin. The unifying mechanism is distance-to-prior minimization: minimal adjustment subject to informational or strategic constraints. Models of inertial updating both recover classical logic and probability theory as special cases and rationalize a wide class of non-Bayesian behaviors observed in human and artificial agents.

Outstanding challenges include:

• Formal comparative analysis of inertial updating across different classes of divergence measures and their behavioral implications for collective dynamics.
• The axiomatization of multi-agent belief inertia in the presence of strategic communication and adversarial environments.
• Empirical calibration of inertia parameters in social networks and organizational settings.
• Development of algorithms robust to inertia-induced failures in online, changing environments.

In sum, the Belief Inertia Argument provides a mathematically rigorous, empirically grounded, and normatively relevant account of how, why, and with what consequences beliefs resist change in individuals, collectivities, and artificial systems.