addendum to obsidian/lesswrong/archived/The Mind-Market Correspondence.html

I have looked through obsidian/shortform/2023%20Natural%20latents.%20I%20think%20his%20“fundamental%20theorem”%20is%20just%20a%20generalization%20of%20[sufficient%20statistics): a statistic \(\Lambda(\mathbf{X})\) that is sufficient for all \(X_1,\dots X_n\) is also sufficient for any “insensitive” variable \(\Lambda'(\mathbf{X})\), because \(\Lambda\) contains every bit of mutual information, and \(\Lambda’\) can’t depend on any \(X_i\)-specific information, so it has to be only determined by mutual information. Furthermore if \(\Lambda\) is both sufficient and insensitive (“a natural latent”), it is a minimal sufficient statistic.

(He doesn’t write these as random functions of \(\mathbf{X}\), but that’s essentially what “resampling latents” does.)

He conjectures that there is a way to naturally construct a natural latent, via “simultaneous resampling”, described in [L2], [L3], [L4].