The Bayesian Origin of the Probability Weighting Function in Human Representation of Probabilities

TL;DR

The study argues that human probability distortion arises from optimal Bayesian decoding of noisy neural representations, formalized as $m = F(p) + \\epsilon$ with posterior decoding yielding $\\hat{p}$ and encoding resources governed by $\\mathcal{J}(p) = (F'(p))^2/\\sigma^2$. It derives that inverse S-shaped probability weighting corresponds to a $U$-shaped allocation of encoding resources and makes testable predictions about prior attraction, variability, and near-optimality, including the Cramér–Rao bound $\\mathrm{MSE} = 1/\\mathcal{J}(p) + O(\\sigma^4)$ as $\\sigma \to 0$. Across Judgment of Relative Frequency, pricing, and choice under risk, the Bayesian framework outperforms bounded log-odds and other descriptive models, with nonparametric priors and nonuniform encodings yielding superior fits and correctly predicting adaptation to bimodal priors. The work provides a normative, unifying account of probability representation that extends to general decision-making and even artificial systems, offering precise predictions about encoding resources and prior–data interactions that can guide future neuroscientific and AI research.

Abstract

Understanding the representation of probability in the human mind has been of great interest to understanding human decision making. Classical paradoxes in decision making suggest that human perception distorts probability magnitudes. Previous accounts postulate a Probability Weighting Function that transforms perceived probabilities; however, its motivation has been debated. Recent work has sought to motivate this function in terms of noisy representations of probabilities in the human mind. Here, we present an account of the Probability Weighting Function grounded in rational inference over optimal decoding from noisy neural encoding of quantities. We show that our model accurately accounts for behavior in a lottery task and a dot counting task. It further accounts for adaptation to a bimodal short-term prior. Taken together, our results provide a unifying account grounding the human representation of probability in rational inference.

Figures (41)

• Figure 1: Distorted probability perception and our Bayesian account. (A) Human decisions systematically distort probabilities, producing the inverse S-shaped probability weighting function central to Prospect Theory. (B) We study a Bayesian encoding–decoding framework: true probabilities $p$ are encoded noisily, combined with a prior, and decoded into perceived probabilities $\hat{p}$. Encoding resources allocated to a probability $p$ are proportional to the slope of the encoding function $F$ and $\sigma$ is the sensory noise standard deviation. This framework explains the origin of the weighting function and predicts that the observed S-shape implies a U-shaped allocation of encoding resources.
• Figure 2: Impact of encoding on bias. (A) U-shaped (blue) and uniform (red) encoding resources $\sqrt{\mathcal{J}(p)}$. (B) The U-shaped encoding generates an S-shaped Likelihood Repulsion bias, $\propto (1/\mathcal{J}(p))'$. (C) The uniform encoding generates a Regression bias $\propto \operatorname{sign}(0.5-p) \frac{A_{1,\sigma}(p)}{\sqrt{\mathcal{J}(p)}}$, maximized at 0,1. Both biases generate overestimation of small and underestimation of large probabilities, but with distinct shapes.
• Figure 3: Results for JRF (Section \ref{['sec:jrf']}). Each Bayesian model variant is defined by a prior and encoding; see text for the shorthands. (A) Model fits to response bias (top) and variability (bottom). Gray curves show human data. The S-shaped bias is only explained by non-uniform encodings (BoundedLOE, FreeE); the variability is only captured by the nonparametric prior (FreeP). Appendix Figures \ref{['fig:jrf-fi-bayesian-perSubject']}–\ref{['fig:jrf-var-perSubject']} provide per-subject results for resources, bias, and variability. (B) Model fit on held-out data (Summed Heldout $\Delta$NLL; lower is better). All bayesian models outperform BLO.
• Figure 4: Resources of Bayesian FreeE FreeP model. IQR denotes interquartile range across subjects.
• Figure 5: Across both pricing (A–B) and choice tasks (C–D), Bayesian models recover U-shaped resources with peaks near 0 and 1, consistent with Prediction 1. In the pricing task, all Bayesian variants outperform BLO, though the fits do not clearly distinguish between U-shaped and uniform encoding (B). In the choice task, however, the freely fitted encoding provides a markedly better fit than a uniform encoding and also outperforms standard parametric weighting functions (C–D).

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3: Repeated from Theorem \ref{['thm:bayesian-bias']}
• proof
• Corollary 4
• proof
• Theorem 5
• proof
• Theorem 6: Optimality of Decoding
• proof