Perceptual Distortions of Probability

• Perceptual distortions of probability are systematic deviations in subjective probability assessments caused by cognitive noise, finite precision, and adaptive nonlinear weighting.
• Models like the probability theory plus noise and quantization frameworks elucidate how memory biases and discrete neural encoding lead to observed under- and overconfidence in risk judgment.
• These insights impact fields such as finance, AI, neuroscience, and robotics by informing decision models, optimizing algorithms, and enhancing interpretation of risk.

Perceptual distortions of probability refer to systematic deviations in subjective probability assessment from objective, mathematically defined probability, as a consequence of limitations, noise, and structural biases in cognitive processing. These distortions are not attributable solely to irrational heuristics; instead, rigorous research has demonstrated that they often arise from fundamentally rational probabilistic computation perturbed by random variability, finite precision, nonlinear weighting functions, and adaptive mechanisms. This entry provides a comprehensive synthesis of core models and empirical findings in the paper of perceptual distortions, integrating perspectives from cognitive psychology, behavioral economics, neuroscience, artificial intelligence, and robotics.

1. Cognitive Noise and Memory Retrieval Biases

A foundational account of perceptual distortions is the "probability theory plus noise" model (Costello et al., 2012). When individuals estimate the probability of an event $A$, they retrieve instances from memory or imagine frequency counts, subject to random noise. If $P(A)$ is the true probability, the perceived estimate $P_e(A)$ follows the transformation:

$P_e(A) = P(A) + d - 2d\,P(A),$

where $d$ is the probability of noise in reading a memory flag. This formulation gives rise to specific, empirically observed biases:

• Conservatism: Low probabilities are biased upward; high probabilities are biased downward, resulting in underconfidence and avoidance of probability extremes.
• Subadditivity: When estimating components of a mutually exclusive event set, the sum of subjective probabilities often exceeds the probability of the union, due to additive noise.
• Conjunction/Disjunction Fallacies: Individual estimates for $A \wedge B$ or $A \vee B$ may, due to noise, violate the monotonicity implied by probability theory, though population-level means remain faithful to the axioms.

Methodologically, these patterns were dissected using composite expressions designed to algebraically cancel noise terms (e.g., $$X_e(A, B) = P_e(A) + P_e(B) - P_e(A \wedge B) - P_e(A \vee B)$$), recovering the underlying normative structure of probability theory in group averages.

2. Quantization and Finite Precision in Neural Encoding

Experimental evidence supports the notion that probability is not represented in the brain as a continuous real number, but instead is discretized through quantization (Tee et al., 2020). The quantized distortion model applies an $n$-bit discretization of continuous probability weighting functions such as Prelec’s:

$$w(x) = \exp\left[ -\delta\,(-\ln x)^\gamma \right], \qquad y = Q_n[w(x)],$$

with $Q_n$ partitioning $[0,1]$ into $2^n$ bins.

Empirical studies using conjunction gambling tasks reveal that the majority (~78%) of participants' probability judgments are best fit by 4-bit models, meaning the brain represents probabilities in only 16 distinguishable categories. This produces "no noticeable difference" (NND) regions where objective probability changes are subjectively invisible and "big noticeable difference" (BND) jumps at bin boundaries. Such quantization underlies significant perceptual distortions in risk assessment, especially in everyday and high-stakes decision contexts.

3. Nonlinear Weighting and Cumulative Prospect Theory

Nonlinear weighting of probability—a key concept in cumulative prospect theory (CPT)—further characterizes perceptual distortions (Liang et al., 2017). Probability weighting functions $w(\cdot)$, strictly increasing and differentiable, transform cumulative probabilities:

$w(p) = v\,p^{\theta+1} + (1-v)[1-(1-p)^{\beta+1}],$

with small probabilities overweighted and moderate/high probabilities underweighted.

In stochastic control formulations, such as continuous-time portfolio optimization, S-shaped utility functions (concave for gains, convex for losses) are combined with probability distortions, altering the effective objective:

$$J(u(\cdot)) = \mathbb{E}\left[ \int_0^T \left\{ S_+(c_t X_t)\, w_+\big(1 - F_{c_t X_t}(c_t X_t)\big) - S_-(c_t X_t)\, w_-\big(1 - F_{c_t X_t}(c_t X_t)\big) \right\} dt + \ell(X_T)\,w'\big(1-F_{X_T}(X_T)\big) \right]$$

This framework clarifies how perceptual distortions—modeled through weighting functions and distinct utility for gains/losses—systematically modify optimal behavior across finance, gambling, and consumption scenarios.

4. Duality with Utility Transforms and Coherence Constraints

A mathematical duality exists between probability distortions and utility transforms (Chambers et al., 2023). Distributional transforms are categorized as:

• Probability Distortion: $T_d(F)(x) = (d \circ F)(x^+)$, where $d$ is an increasing map on $[0,1]$.
• Utility Transform: $T_u(F) = F \circ u^{-1}$, where $u$ is a strictly increasing utility function.

Key results show that probability distortions commute with all utility transforms, and vice versa. Rank-dependent utility is characterized by compositional commutation properties:

$\text{RDU}(F) = \int u(x)\,d(d \circ F)(x)$

This unifies classic behavioral theories: expected utility (EU), dual utility (DU), and rank-dependent utility (RDU), providing a rigorous taxonomy for distorted probability perception.

Further, distortion coherence (Chambers et al., 2023) imposes the requirement that the order of conditioning and distortion commute:

$\phi(p(\cdot|E)) = [\phi(p)](\cdot|E)$

Under these constraints, admissible distortions must take the power–weighted form:

$$\phi(p)(\omega) = \frac{\psi(\omega)\,p(\omega)^\alpha}{\sum_{\omega'} \psi(\omega')\,p(\omega')^\alpha}$$

This structure generalizes to signals and connects to motivated beliefs and non-EU models, including those explaining the Allais paradox and base-rate neglect.

5. Perceptual Distances and Multisource Cost Measures

Beyond direct probability estimation, perceptual distances influence the cost and granularity of learning and discrimination (Walker-Jones, 2019). The Multisource Shannon Entropy (MSSE) measure augments classical Shannon entropy with distance-based multipliers:

$$C(\mathcal{P}, \mu) = \lambda(\mathcal{P})\, \mathcal{H}(\mathcal{P}, \mu) = \lambda(\mathcal{P}) \left(-\sum_{i=1}^m \mu(A_i)\log\mu(A_i)\right)$$

When partitions have varying perceptual distances ($\lambda$), differentiation among events with higher intrinsic similarity is more costly, leading to smooth and context-sensitive distortions in choice probabilities:

$$\Pr(n | \omega) = \frac{[\Pr(n)^{\lambda_1/\lambda_M} \cdots ] \cdot \exp[v_n(\omega)/\lambda_M]}{\sum_\nu [\Pr(\nu)^{\lambda_1/\lambda_M} \cdots ] \cdot \exp[v_\nu(\omega)/\lambda_M]}$$

A plausible implication is that informational bias arises naturally when perceptual distances are heterogeneous, impacting both welfare analysis and econometric modeling.

6. Perceptual Effects in Machine Learning and Artificial Systems

Perceptual distortions manifest in artificial systems as well. In deep learning, surrogate explainers for black-box image classifiers generate varying local explanations depending on perceptual distortions of input data—even when probability estimates remain unchanged (Hepburn et al., 2021). Robustness is enhanced by weighting sample neighborhoods using perceptual metrics such as MS-SSIM or NLPD, yielding more coherent explanations that remain stable across noise and compression artifacts.

Robotic agents using imperfect sensors experience perceptual distortions in their environmental representation (Warutumo et al., 10 Jul 2025). Sensor mappings $s = T(p) + \varepsilon$ create warped perceptual spaces, evidenced by non-Euclidean sensor clusters and emergent structures through unsupervised learning. The probabilistic belief the robot forms about its environment reflects these distortions yet remains functional due to adaptation and clustering.

7. Perceptual Biases in Social Forecasting and Human-Centric Model Alignment

Judgment under uncertainty is susceptible to perceptual conflation between probability forecasts and tail risk (Taleb et al., 2023). Expert probability assessments ($P_K$) of extreme events are thin-tailed and bounded, whereas tail expectation ($G_K = \int_K^\infty g(x)f(x)\,dx$) under fat-tailed distributions exhibits explosive sensitivity. The mathematical non-equivalence ($K = L(P_K)^{-1/\alpha}$) means small errors in $P_K$ yield disproportionate risk misassessment, challenging the adequacy of forecasting tournaments.

In AI alignment, perceptual biases grounded in prospect theory are exploited to optimize generative model training (Liu et al., 29 Sep 2025). Human-perceived probability is modeled by a value function $v(z;\lambda,\alpha,z_0)$ and a capacity function $\Omega^+(a;\gamma)$ that overweights extreme events. Policy gradient clipping (in PPO/GRPO) operationalizes these distortions as perceptual losses, resulting in humanline alignment schemes that synchronize the reference model and asymmetrically clip likelihood ratios:

$$\frac{\pi_\theta(y_t|x,y_{<t})}{\pi_\mathrm{ref}(y_t|x,y_{<t})} < M' \cdot B,\quad B \sim \mathrm{Beta}(\gamma, 1)$$

Empirical results show that offline training with humanline clipping matches the performance of online alignment, demonstrating the advantage of explicitly modeling perceptual distortions in utility optimization.

Summary Table: Major Mechanisms of Perceptual Probability Distortion

Mechanism	Formalism/Process	Key Empirical Consequence
Memory Noise	$P_e(A) = P(A) + d - 2d\,P(A)$	Conservatism, fallacies
Quantization	$y = Q_n[w(x)]$	Discrete probability bins
Nonlinear Weighting (CPT)	$w(\cdot)$, S-shaped functions	Overweighting rare events
Power-Weighted Distortion	$\phi(p)(\omega)$ formula	Allais paradox, base-rate neglect
Perceptual Cost (MSSE)	$$C(\mathcal{P},\mu) = \lambda(\mathcal{P})\,\mathcal{H}$$	Informational bias

Concluding Remarks

Perceptual distortions of probability are mathematically inevitable in any system—biological or artificial—subject to noise, finite precision, nonlinear transformation, cost constraints, and adaptive structuring. These distortions do not imply irrationality or mere heuristic processing; rather, in human and artificial agents, they often reflect optimal or constrained computation under resource limitations and environment-specific perceptual metrics. Future work may further elaborate on how early perceptual noise, graded event membership, or conditional probability estimation interact to shape probability judgment in complex, real-world environments.