Tactile spatiotemporal illusions have long intrigued and puzzled researchers. Perhaps the earliest description was made by Weber, who in 1834 reported that the perceived separation between two fixed caliper points expands as the points are dragged along the skin from the forearm towards the fingertips 
. Weber concluded, in agreement with modern studies 
, that distance is underestimated on skin regions with poor tactile acuity, a phenomenon termed spatial compression
by Green 
. Some 100 years after Weber's publication, Helson 
described the tau effect, showing that perceived tactile distance depends on inter-stimulus timing. The rabbit illusion later described by Geldard and Sherrick 
confirmed the temporal dependence of spatial perception, while the kappa effect, described concurrently by Cohen and colleagues 
in vision and Suto 
in touch, revealed the spatial dependence of temporal perception.
Several clever theoretical explanations have been advanced to account for these illusions. Collyer 
proposed that the brain expects movement to occur at the same velocity in all segments of a multi-segment stimulus sequence, and that it adjusts space and time perception accordingly. For instance, the classic tau effect () was hypothesized to arise because the brain expects movement to occur at the same velocity between the first and second, as between the second and third stimulus positions. A related line of reasoning was followed by Jones and Huang 
, who modeled perceived inter-stimulus distance and time as weighted averages of actual and expected inter-stimulus distance and time, with the expected values derived from a constant velocity assumption. A different and particularly creative approach was taken by Brigner, who hypothesized that spatiotemporal illusions result from rotation of a perceptual space-time coordinate frame 
. The hypothesized transformation achieves spatial and temporal perceptual adjustments in a way that is, roughly, the converse of that shown in The trajectory line (filled circles) remains fixed, while the space and time axes rotate together counterclockwise.
None of these interesting explanations has been applied quantitatively to a wide variety of experimental data, and each has shortcomings. Collyer's hypothesis may prove relevant to the perception of sequences with three or more stimulus locations, but its application to sequences with just two spatial positions, which also produce illusions (e.g., ), is less clear. The weighted average model proposed by Jones and Wang leaves unanswered the question of how the relative weights are determined, and particularly what mechanism governs their evident dependence on the duration of the stimulus sequence. Brigner's intriguing proposal is able to explain, at least qualitatively, perceptual illusions evoked by stimuli at just two positions, but how or why the brain would undertake the proposed coordinate transformation is unclear.
The Bayesian observer model described here provides a coherent explanation for perceptual length contraction and time dilation, and replicates the rabbit illusion, the tau effect, the kappa effect, and a variety of other spatiotemporal illusions. The results suggest that the brain takes advantage of the expectation for slow speed, presumably based in tactile experience, to improve perception beyond the limits imposed by spatial and temporal uncertainty inherent in the sensorineural signal.
The Bayesian observer's slow-speed expectation recalls a visual model with that expectation that reproduces contrast effects on motion perception 
. The remarkable explanatory power of these models supports Helmholtz's view of perception as a process of unconscious inference, in which “previous experiences act in conjunction with present sensations to produce a perceptual image” 
. The perceptual space-time distortions that emerge from the Bayesian observer, and characterize human tactile perception, are loosely analogous to the physical length contraction and time dilation described in the Special Theory of Relativity 
. I do not attach special significance to this analogy, but note simply that it arises because any postulated constraint on speed naturally yields distortions of space and/or time.
The Bayesian observer makes several novel testable predictions and suggests many experiments. For example, the model predicts more pronounced time dilation (), as well as less pronounced length contraction (), on body areas with finer tactile acuity, and it predicts a perceptual speed limit on the velocity evoked by dual punctate stimuli with fixed spacing (). Temporal perception experiments will determine whether the kappa effect is indeed more pronounced on body areas with finer tactile acuity (), while velocity perception experiments will provide data for comparison to the curves shown in . In addition, the model suggests experiments with within-subjects designs to determine the contributions of σs and σv to variation in λ, not only across body regions (), but also across perceptual tasks and as a result of perceptual learning. Finally, although designed to model tactile perception, the Bayesian observer may prove relevant to perception in other sensory modalities that show similar spatiotemporal illusions. For instance, , translated to visual perception, predicts a greater kappa effect for foveal than peripheral stimulus sequences.
Important work related to the model remains to be done. Experiments are needed to determine the precise shapes of the prior and likelihood distributions assumed by human observers as they perceive tactile stimulus sequences, as has been done for visual motion perception 
. The Gaussian priors and likelihoods used in the model may need to be refined as a result of such experiments. Furthermore, theoretical work is needed to extend the model to treat the perception of more complex punctate stimulus sequences (e.g., 
), and of smoothly moving objects 
. Interestingly, humans progressively underestimate the fixed distance traversed by a brush swept briskly across the skin as sweep duration decreases 
, a result in qualitative agreement with Equation 1.
Finally, research is needed to determine where in the brain the Bayesian probability distributions hypothesized to serve tactile perception are represented, and by what neural mechanism they are generated. Interestingly, topographically appropriate somatosensory cortical activity accompanies illusory rabbit percepts on the forearm 
. Research is needed, then, to explore connections between models of somatosensory cortical function recently proposed to account for the rabbit illusion 
, and hypothesized neural representations of Bayesian probability distributions