Figure 2.

Bayesian model. (A) The observer’s likelihood function, prior probability density, and posterior probability density in response to taps sensed (i.e., measured by the observer) at positions (x_1m, x_2m) = (3, 7 cm) (open red circles in all plots). Each pixel in the intensity plots represents a candidate trajectory: a possible tap 1 position and tap 2 position pair (x₁, x₂). Lighter color indicates higher probability (each plot is individually auto-scaled to take advantage of the full brightness range). The measured trajectory length is l_m = x_2m − x_1m = 4 cm. Top: the observer’s likelihood function plots the probability of the measured trajectory given each candidate trajectory. The observer understands that a single tap at any location produces a measurement drawn from a Gaussian distribution centered at that location, with standard deviation σ_s; thus, the likelihood function is a two-dimensional Gaussian density centered on the measured trajectory. Middle: the observer expects slow movement to occur more commonly; we model this expectation as a Gaussian distribution over trajectory speed, with mean zero and standard deviation, σ_v. Consequently, the observer expects closely spaced taps, and its prior is maximal along the x₁ = x₂ diagonal. Bottom: the posterior probability of each trajectory is proportional to the product of its likelihood and prior. The mode of the posterior (filled red circle) is the percept. (B) Space-time plots equivalently illustrate the inference process. Top: open red circles show measured tap positions (vertical-axis) and times of occurrence (horizontal-axis). Error bars (±1σ_s) represent the spatial imprecision of the measurements. The slope of the line connecting the taps is the measured trajectory speed: l_m /t = 4 cm/0.15 s = 27 cm/s. Middle: the observer’s low-speed expectation is represented by the line of slope zero and diagonal lines of slopes ±1σ_v = ±10 cm/s. The distance traversed at speed σ_v in time t is tσ_v = 1.5 cm. The ascending diagonal line is shallower than the measured velocity: 10 cm/s < 27 cm/s. Equivalently, tσ_v = 1.5 cm < l_m = 4 cm. Thus, the measured trajectory violates the observer’s low-speed expectation. Bottom: the perceived trajectory (filled red circles and red line) is a compromise between the measured trajectory (open circles, reproduced from top panel) and expectation (middle panel). Each tap has migrated perceptually by 1 cm toward the other, resulting in perceptual length contraction: l* = 2cm < l_m = 4 cm. The perceived trajectory speed is l*/t = 2 cm/0.15 s = 13 cm/s. In both panels, σ_s = 1 cm, σ_v = 10 cm/s, t = 0.15 s, x_1m = 3 cm, x_2m = 7 cm.