“$500 + $500 Bounty Problem: An (Approximately) Deterministic Maximal Redund Always Exists” by johnswentworth, David Lorell
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Audio note: this article contains 61 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

A lot of our work involves "redunds". A random variable <span>Gamma</span> is a(n exact) redund over two random variables <span>X_1, X_2</span> exactly when both

<span>X_1 rightarrow X_2 rightarrow Gamma</span>

<span>X_2 rightarrow X_1 rightarrow Gamma</span>

Conceptually, these two diagrams say that <span>X_1</span> gives exactly the same information about <span>Gamma</span> as all of <span>X</span>, and <span>X_2</span> gives exactly the same information about <span>Gamma</span> as all of <span>X</span>; whatever information <span>X</span> contains about <span>Gamma</span> is redundantly represented in <span>X_1</span> and <span>X_2</span>. Unpacking the diagrammatic notation and simplifying a little, the diagrams say <span>P[Gamma|X_1] = P[Gamma|X_2] = P[Gamma|X]</span> for all <span>X</span> such that <span>P[X] > 0</span>.

The exact redundancy conditions are too restrictive to be of much practical relevance, but we are [...]

Outline:

(02:31) What We Want For The Bounty

(04:29) Some Intuition From The Exact Case

(05:57) Why We Want This

First published: May 6th, 2025

Source: https://www.lesswrong.com/posts/sCNdkuio62Fi9qQZK/usd500-usd500-bounty-problem-an-approximately-deterministic)

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