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Scale invariance in cognition

Nick Chater Department of Psychology University College London n.chater@ucl.ac.uk. Scale invariance in cognition. What is scale-invariance?. In a nutshell: Throw away “units” Can you reconstruct them from your data? If not , phenomenon is scale-invariant.

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Scale invariance in cognition

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  1. Nick Chater Department of Psychology University College London n.chater@ucl.ac.uk Scale invariance in cognition

  2. What is scale-invariance? • In a nutshell: • Throw away “units” • Can you reconstruct them from your data? • If not, phenomenon is scale-invariant Only power laws yx are scale invariant

  3. Fractals With consequent power laws E.g., no. branches vs. branch diameter The ubiquity of scale-invariance I From West & Brown, JEB, 05

  4. metabolic power vs body mass (Kleiber, 1932) The ubiquity of scale-invariance II From West & Brown, JEB, 05

  5. Scale-invariance as a “null hypothesis” for the cognitive and brain sciences This null hypothesis implies many well-known psychological laws… City sizes Sizes of firms River sizes Earthquakes Distribution of digits (Benford’s Law) Word frequencies (Zipf’s Law) Mandelbrot: Scale-invariance as a primitive The ubiquity of scale-invariance III

  6. Location parameter: invariance under translation fm(x)=f(x-m) Mean of a Gaussian Direction of a sound source Simple default: Uniform prior Scale parameter: invariance under change of scale fs(x)=f(x/s)/s Variance (of zero-mean) Gaussian Sound pressure Frequency Simple default: Uniform prior in log space An Aside: Scale-invariance and non-informative priors Improper priors can require careful handling; and there are many proposals about how priors should be set

  7. From scale-invariance to psychological “laws”

  8. Weber’s law Endless cases of invariance, in perception, motor control, learning and memory

  9. Herrnstein’s matching law • Probability of choosing an option, Ri based on Payoff(Ri) • Attractiveness of an option must be a power function (Payoff(Ri))α • Choice probability can only depend on ratios of attractiveness: A very widely observed law of behavior (and cf Luce’s Choice Rule)

  10. Serial position in immediate free recall Data from Murdock, 1962; model fits using SIMPLE (Brown, Neath & Chater)

  11. Confusion in memory for serial order(data fits using SIMPLE)

  12. Memory retrieval over different time periods in retrospective memory(Maylor, Chater & Brown, 2001, PB&R)

  13. And prospective memory

  14. Violations may be informative; conformity is not • 1. Scale in the environment • Non-invariance of speech • Bat auditory cortex • 2. Where there is a transition between “dominant” processes, may expect scale non-invariance • But not, perhaps, STM vs LTM

  15. Analyse time series of errors on successive trials Log power vs log freq 1/f noise + white noise for high frequencies Roughly: 1/f noise in internal clock + motor jitter Gilden et al (1995): Replicating time-intervals Data Simulation

  16. Conclusion • Scale-invariance as a null hypothesis for cognitive science • Explains a high proportion of psychological “laws” (!) • Violations may be informative • Scale-invariance as a building block for building “default” cognitive models • And might be especially interesting, when combined with other building blocks, e.g., Bayes, simplicity, invertability, seriality

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