A Bayesian Log-Time State-Space Clock for Human Time Reproduction
This work tests whether a compact Bayesian state-space mechanism in log-time can jointly explain mean reproduction behaviour, central tendency, multiplicative error structure, and duration-dependent variability—not merely whether it can achieve a good average fit.
15,264
Clean trials
0.8237 s
RMSE
0.5833 s
MAE
0.7099
R²
Model architecture
Sequential inference in log-time
On each trial, an effective target duration and a covariate-dependent multiplicative gain define an internal log-time signal. The latent perceived-duration state is updated sequentially with Kalman-like weighting, and observed reproductions are represented with a lognormal observation model.
Parameters were estimated with constrained multi-start optimization, using maximum likelihood or MAP when weak priors were enabled to stabilize weakly identified parameters. Evaluation then used posterior-predictive summaries and model-fit diagnostics rather than treating a single objective value as sufficient evidence.
Dataset and fit
Strong mean-level agreement with limited data exclusion
The raw dataset contained 15,336 trials. Pre-specified validity filters retained 15,264 trials and removed 72 (0.47%). In the seconds domain, the model achieved RMSE = 0.8237 s, MAE = 0.5833 s, and R² = 0.7099. In log space, RMSE was 0.2490 and MAE was 0.1862.
Mean predictions reproduced the characteristic central-tendency pattern: shorter intervals were over-reproduced on average, longer intervals were under-reproduced, and the crossover occurred in the mid-range of the target set.
Posterior-predictive diagnosis
The strongest result is also a model limitation
Empirical coefficient of variation decreased with duration, approximately from 0.29 to 0.20, while posterior-predictive coefficient of variation under the stationary model remained comparatively stable around 0.24. Residuals also showed clustering and burst-like deviations rather than fully independent stationary noise.
The model therefore captures central tendency and much of the mean-level structure, but its stationary noise formulation does not explain the complete duration-wise variability pattern. This mismatch is treated as a scientific diagnostic, not hidden by the overall R².
Interpretation
A useful model because its failure mode is visible
The paper supports a Bayesian log-time account as a compact explanation of important features of human time reproduction in this dataset, while also showing where that account is incomplete. The central scientific point is not that the model is universally sufficient, but that explicit posterior-predictive checking reveals which behavioural regularities require richer variability structure.