Audio note: this article contains 94 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Introduction

In this short post we'll discuss fine-grained variants of the law of large numbers beyond the central limit theorem. In particular we'll introduce cumulants as a crucial (and very nice) invariant of probability distributions to track. We'll also briefly discusses parallels with physics. This post should be interesting on its own, but the reason I'm writing it is that this story contains a central idea for (one point of view) on a certain exciting physics-inspired point of view on neural nets. While the point of view has so far been explained in somewhat sophisticated physics language (involving quantum fields and Feynman diagrams), the main points can be explained without any physics background, purely in terms of statistics. [...]

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Outline:

(00:16) Introduction

(01:17) Review of the central limit theorem (as a law of large numbers correction)

(01:24) The law of large numbers

(04:09) The central limit theorem

(06:41) The third law and beyond

(13:00) Higher cumulants and higher laws

(15:33) Multiple random variables

(16:55) Connections to neural nets

(18:51) Connections to physics and the stationary phase formula

The original text contained 3 footnotes which were omitted from this narration.

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First published:
January 4th, 2025

Source:
https://www.lesswrong.com/posts/EhTMM77iKBTBxBKRe/the-laws-of-large-numbers

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Narrated by TYPE III AUDIO.

“The Laws of Large Numbers” by Dmitry Vaintrob

LessWrong (30+ Karma)