Assessing the Perceived Predictability of Functions

How do we perceive the predictability of functions? We derive a rational measure of a function’s predictability based on Gaussian process learning curves. Using this measure, we show that the smoothness of a function can be more important to predictability judgments than the variance of additive noise or the number of samples. These patterns can be captured well by the learning curve for Gaussian process regression, which in turn crucially depends on the eigenvalue spectrum of the covariance function. Using approximate bounds on the learning curve, we model participants’ predictability judgments about sampled functions and find that smoothness is indeed a better predictor for perceived predictability than both the variance and the sample size. This means that it can sometimes be preferable to learn about noisy but smooth functions instead of deterministic complex ones.