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The tau effect is a spatial perceptual illusion that arises when observers judge the distance between consecutive stimuli in a stimulus sequence. When the distance from one stimulus to the next is constant, and the time elapsed from one stimulus to the next is also constant, subjects tend to judge the distances, correctly, as equal. However, if the distance from one stimulus to the next is constant, but the time elapsed from one stimulus to the next is not constant, then subjects tend to misperceive the interval that has the shorter temporal interval as also having a shorter spatial interval. Thus, the tau effect reveals that stimulus timing affects the perception of stimulus spacing. Time is also a perceived quantity and subject to its own illusions; research indicates that in the tau effect, perceived stimulus spacing follows perceived (phenomenal) time rather than actual (physical) time.
Unlike the constant velocity hypothesis, the Bayesian model replicates the underestimation in perceived distance that occurs even when only two stimuli are presented in rapid succession. For the case of two taps to the skin, the Bayesian model perceives the length between taps, l*, to be a function of the actual length, l, and the elapsed time, t:
The parameter tau (τ) is proportional to the observer's spatial uncertainty (specifically, it is the spatial standard deviation divided by the low-speed prior standard deviation). Consistent with this model, Tong et al. (2016) showed that stimulus pairs consisting of weaker taps, which are localized with greater uncertainty than stronger taps, result in more pronounced length contraction. Modeling the tau effect that occurs in the perception of 3-tap sequences, Goldreich and Tong (2013) compared the Bayesian model with a low-speed expectation to a Bayesian model with a low-acceleration expectation — similar to the constant-velocity hypothesis. They found that the low-speed prior model provided better fits to the human tactile tau effect data. When time is inaccurately perceived (i.e., because of the kappa effect), the Bayesian observer model judges stimulus spacing to follow perceived time rather than actual time, consistent with reports from human subjects.
If observers interpret rapid stimulus sequences in light of an expectation regarding velocity, then it would be expected that not only spatial, but also temporal illusions would result. This indeed occurs in the kappa effect: When the temporal separation between stimuli is constant and the spatial separation is varied, the observer's temporal interval judgment is influenced by the spatial distance between consecutive stimuli. Specifically, longer spatial intervals are perceived to occupy longer temporal intervals. The kappa effect is therefore the temporal perceptual analog of the tau effect.
Goldreich (2007) linked the tau, rabbit, and kappa effects to the same underlying expectation regarding movement speed. He noted that, when stimuli move rapidly across space, "perception strikingly shrinks the intervening distance, and expands the elapsed time, between consecutive events". Goldreich (2007) termed these two fundamental perceptual distortions "perceptual length contraction" (tau effect, rabbit illusion) and "perceptual time dilation" (kappa effect) in analogy with the physical length contraction and time dilation of the theory of relativity. Perceptual length contraction and perceptual time dilation result from the same Bayesian observer model, one that expects stimuli to move slowly. Analogously, in the theory of relativity, length contraction and time dilation both occur when a physical speed (the speed of light) cannot be exceeded.
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