The origin of Vierordt's law: The experimental protocol matters

Abstract

In 1868, Karl Vierordt discovered one type of errors in time perception-an overestimation of short duration and underestimation of long durations, known as Vierordt's law. Here we reviewed the original study in its historical context and asked whether Vierordt's law is a result of an unnatural experimental randomization protocol. Using iterative Bayesian updating, we simulated the original results with high accuracy. Importantly, the model also predicted that a slowly changing random-walk sequence produces less central tendency than a random sequence with the same durations. This was validated by a duration reproduction experiment from two sequences (random and random walk) with the same sampled distribution. The results showed that trial-wise variation influenced the magnitude of Vierordt's law. We concluded that Vierordt's law is caused by an unnatural yet widely used experimental protocol.

Individual beliefs about temporal continuity explain variation of perceptual biases

“…According to the model, for these participants, the overall central tendency bias should be smaller, if the stimuli are indeed similar from trial to trial. This was validated by showing in our previous study (Glasauer & Shi 2021), that the central tendency in sequences with complete random stimulus order was larger than in sequences with random-walk fluctuation. Here we showed that this decrease in central tendency and, more importantly, the remaining central tendency, is well-predicted by the two-state model.…”

“…2C, red time course, for an example of such a random walk). As explained in our previous paper (Glasauer & Shi 2021), this condition tests the prediction of the simple iterative and the Petzschner & Glasauer (2011) explanatory models, which both predict that the central tendency vanishes in the random walk condition. However, while it was found that the central tendency indeed decreased substantially and was significantly smaller during random walk (t-test, n=14, p<0.0001; see Fig.…”

“…However, while it was found that the central tendency indeed decreased substantially and was significantly smaller during random walk (t-test, n=14, p<0.0001; see Fig. 7A), it did not completely vanish and was still larger than predicted by these previous models (Glasauer & Shi 2021). This was, however, not true for all subjects: for some subjects, the central tendency was no longer different from zero (see example data in Fig.…”

“…Investigating sequential dependence can thus not only help distinguishing between the two cases -sequential dependence is only predicted if temporal continuity is assumed -but also help resolving whether individual differences are due to different amounts of sensory or prior variability, or due to different believes about the generative process of stimuli in the world. In the following we thus examined central tendency and serial dependence in three experimental data sets of magnitude estimation: one from duration reproduction (Glasauer & Shi 2019, 2021, and two from linear and angular distance reproduction (Petzschner & Glasauer 2011, data set published as Petzschner & Glasauer 2020).…”

Conceptually plausible Bayesian inference in interval timing

“…Although it is not straightforward to empirically dissociate linear versus logarithmic encoding of time, some psychophysical [32][33][34] and neurophysiological evidence [35] suggests a logarithmic encoding of time. Moreover, several Bayesian models of time estimation have assumed that Bayesian integration takes place on a logarithmic scale [36][37][38], which naturally accounts for lognormally distributed time estimates and the scalar property. The MLN model departs from these logarithm-based Bayesian models in two theoretically important ways: (i) assumptions about the scalar property and (ii) the scale at which Bayesian integration is applied: logarithmic or linear.…”