Oct

23

 Mumford & Desolneux's book Pattern Theory: The Stochastic Analysis of Real-World Signals (Applying Mathematics) is the most promising work in foundations of statistical modelling I think I've ever read. The main idea is that whatever you're trying to model is some infinite-dimensional parameter space. The shape of the neighbourhoods and distances between them then gives structure to the randomness.

(My go-to example of infinite-dimensional space is handwriting: those who have tried to classify it, for example Douglas Hofstadter and Donald Knuth, have not been able to generate all possible ways to write eg "lowercase a" with any number of knobs. This is consistent with surfaces having a greater cardinality than reals. Demo of handwriting variation in a finite basis.

This is the rare book that doesn't get overwhelmed by the grandiosity of its own mathematical techniques. Lie theory and wavelets are just foundations, not the "Reverse Kolmogorov Smirnov Filter" of Nuclear Phynance.

P.S (Colour and music are also infinite-dimensional, in theory, as waveforms at least. Humans cannot resolve infinitesimal frequencies by eye or ear, of course. And ħ probably means that space can't resolve infinitesimal frequencies either. But theoretical waveforms are simpler; as long as a super high pitch was super soft, or conversely a very low pitch was very loud, the energy could be finite and constant.)

Here are some more great articles on loud low pitches:

"Strange but true: black holes sing"

"Have You Heard About B-Flat?"

"Big Long Cycle = Trend"


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