Poster abstract

Everyday magnitude judgments are context-dependent: a large rat may still be smaller than a small cat. Such judgments require transforming an absolute magnitude (θ) into its relative position (ρ) within a context (c). This flexible transformation is formalized as: ρ = F_c(θ), where F_c is the context’s cumulative distribution function (CDF). Perception, however, yields an uncertain posterior belief p(θ|m) rather than an exact value of θ. Whether this uncertainty is discarded or preserved during the transformation remains unknown.

We contrast two hypotheses regarding the order of averaging and transformation. Under the Point-estimate hypothesis (H₁), the posterior collapses into a single estimate before mapping, predicting F_c(𝔼[θ|m]). Under the Distributional hypothesis (H₂), the full posterior is transformed before averaging, predicting 𝔼[F_c(θ)|m]. Although transforming a full distribution is computationally more demanding, H₂ offers a normative advantage: it is Bayes-optimal under L₂ loss, minimizing the expected squared error of estimating relative magnitude.

Because natural magnitude distributions are non-uniform, F_c is inherently nonlinear. By Jensen’s inequality, H₂ predicts higher relative estimates than H₁ in convex regions of F_c and lower estimates in concave regions. To test these predictions, we designed a psychophysical task where participants either reproduced absolute durations or reported context-relative positions of the occluded duration of a rotating bar across temporal contexts with opposing CDF curvature profiles. Behavioral data exhibited systematic, curvature-dependent shifts predicted by H₂. These findings support the hypothesis that the brain preserves and propagates perceptual uncertainty when transforming absolute magnitudes into context-relative coordinates.