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Keywords:
expectation maximization problem; EMX; continuum hypothesis; independence of ZFC; measurability
expectation maximization problem; EMX; continuum hypothesis; independence of ZFC; measurability
Summary:
It was shown that there is a statistical learning problem -- a version of the expectation maximization (EMX) problem -- whose consistency in a domain of cardinality continuum under the family of purely atomic probability measures and with finite hypotheses is equivalent to a version of the continuum hypothesis, and thus independent of ZFC. K. P. Hart had subsequently proved that no solution to the EMX problem can be Borel measurable with regard to an uncountable standard Borel structure on $X$, and so the independence result could just be an artefact of a model allowing non-measurable learning rules. In this note we reinforce the point somewhat by observing that such a solution cannot even be Lebesgue measurable.
References:
[1] Ben-David S., Hrubeš P., Moran S., Shpilka A., Yehudayoff A.: A learning problem that is independent of the set theory ZFC axioms. available at arXiv 1711.05195v1 [cs.LG] (2017), 17 pages.
[2] Ben-David S., Hrubeš P., Moran S., Shpilka A., Yehudayoff A.: Learnability can be undecidable. Nat. Mach. Intell. 1 (2019), no. 1, 44–48. DOI 10.1038/s42256-018-0002-3 | MR 4233057
[3] Hart K. P.: Machine learning and the continuum hypothesis. Nieuw Arch. Wiskd. (5) 20 (2019), no. 3, 214–217. MR 4448792
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