# Assessing the Perceived Predictability of Functions

- Eric Schulz,
*University College London*
- Josh Tenenbaum,
*Massachusetts Institute of Technology, Cambridge, MA, USA*
- David Reshef,
*Massachusetts Institute of Technology, Cambridge, MA, USA*
- Maarten Speekenbrink,
*University College London*
- Samuel Gershman,
*Massachusetts Institute of Technology, Cambridge, MA, USA*

## Abstract

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.

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