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Multimedia Battery for Assessment of Cognitive and Basic Skills in Mathematics (BM-PROMA)

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Power laws from individual differences in learning and forgetting: mathematical analyses.

Jaap M J Murre¹, Antonio G Chessa

¹University of Amsterdam, Amsterdam, The Netherlands. jaap@murre.com

Psychonomic Bulletin & Review

|April 7, 2011

Summary

Power laws in learning and forgetting may emerge from averaging simpler exponential functions. Mathematical proofs and simulations confirm this, especially with specific learning rate distributions.

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Area of Science:

Cognitive Science
Mathematical Psychology
Learning Sciences

Background:

The Power Law of Learning describes performance improvement with practice.
Forgetting may also follow a power law.
Previous simulations suggested power laws can arise from averaging non-power functions.

Purpose of the Study:

To mathematically prove how power functions emerge from averaging exponential functions.
To investigate the role of learning rate distributions in this phenomenon.
To assess the practical implications for empirical learning data.

Main Methods:

Mathematical derivation of power function emergence.
Extensive computer simulations.
Analysis of averaging exponential functions with specific learning rate distributions (gamma, uniform, half-normal).

Main Results:

Mathematical proof confirms power functions emerge from averaging exponential functions under specific distributions.
Simulations support the theoretical findings.
The findings demonstrate a mechanism for power law emergence beyond inherent power-law processes.

Conclusions:

Power laws in learning and forgetting are not necessarily fundamental but can emerge from underlying exponential processes.
The distribution of learning rates significantly influences the emergence of power law behavior.
These findings have implications for interpreting empirical learning and forgetting data.