Interesting Esoterica

One parameter is always enough

Article by Steven T. Piantadosi
  • Published in 2018
  • Added on
We construct an elementary equation with a single real valued parameter that is capable of fitting any “scatter plot” on any number of points to within a fixed precision. Specifically, given given a fixed \(\epsilon \gt 0\), we may construct \(f_\theta\) so that for any collection of ordered pairs \( \{(x_j,y_j)\}_{j=0}^n \) with \(n,x_j \in \mathbb{N}\) and \(y_j \in (0,1)\), there exists a \(\theta \in [0,1]\) giving \(|f_\theta(x_j)-y_j| \lt \epsilon\) for all \(j\) simultaneously. To achieve this, we apply prior results about the logistic map, an iterated map in dynamical systems theory that can be solved exactly. The existence of an equation \(f_\theta\) with this property highlights that “parameter counting” fails as a measure of model complexity when the class of models under consideration is only slightly broad.

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BibTeX entry

@article{OneParameterIsAlwaysEnough,
	title = {One parameter is always enough},
	abstract = {We construct an elementary equation with a single real valued parameter that is capable of fitting any “scatter plot” on any number of points to within a fixed precision.  Specifically, given given a fixed \(\epsilon \gt 0\), we may construct \(f{\_}\theta\) so that for any collection of ordered pairs \( \{\{}(x{\_}j,y{\_}j)\{\}}{\_}{\{}j=0{\}}^n \) with \(n,x{\_}j \in \mathbb{\{}N{\}}\) and \(y{\_}j \in (0,1)\), there exists a \(\theta \in [0,1]\) giving \(|f{\_}\theta(x{\_}j)-y{\_}j| \lt \epsilon\) for all \(j\) simultaneously. To achieve this, we apply prior results about the logistic map, an iterated map in dynamical systems theory that can be solved exactly.  The existence of an equation \(f{\_}\theta\) with this property highlights that “parameter counting” fails as a measure of
model complexity when the class of models under consideration is only slightly broad.},
	url = {https://colala.bcs.rochester.edu/papers/piantadosi2018one.pdf},
	year = 2018,
	author = {Steven T. Piantadosi},
	comment = {},
	urldate = {2018-06-06},
	collections = {Attention-grabbing titles,Probability and statistics,Fun maths facts}
}