$500 + $500 Bounty Problem: An (Approximately) Deterministic Maximal Redund Always Exists
Published on May 6, 2025 11:05 PM GMTA lot of our work involves "redunds".[1] A random variable Γ is a(n exact) redund over two random variables X1,X2 exactly when bothX1→X2→ΓX2→X1→ΓConceptually, these two diagrams say that X1 gives exactly the same information about Γ as all of X, and X2 gives exactly the same information about Γ as all of X; whatever information X contains about Γ is redundantly represented in X1 and X2. Unpacking the diagrammatic notation and simplifying a little, the diagrams say P[Γ|X1]=P[Γ|X2]=P[Γ|X] for all X such that P[X]>0.The exact redundancy conditions are too restrictive to be of much practical relevance, but we are more interested in approximate redunds. Approximate redunds are defined by https://www.lesswrong.com/posts/XHtygebvHoJSSeNPP/some-rules-for-an-algebra-of-bayes-nets