Thursday, July 30, 2009

#70. New paper by Grigorii Melnichenko

Stop the press, here comes an unexpected paper!

JMVA is going to publish yet another fuzzy paper; particularly, a paper on quantum logic.

Grigorii Melnichenko. Energy discriminant analysis, quantum logic, and fuzzy sets. Journal of Multivariate Analysis, to appear. Update: JMVA 101 (2010), 68-76.

As you may or may not know, broadly speaking there are two main approaches to what a physicist might call non-Kolmogorovian probability. One uses operator-valued probabilities and has led to a quite fruitful field of research called non-commutative or free probability. The other goes back to work by Birkhoff and von Neumann in the 30's, and uses real-valued probabilities but defined on structures poorer than sigma-algebras (e.g. orthomodular lattices; in the original work, events are defined as closed subspaces of a Hilbert space).

While the first-mentioned approach has found strong supporters among statisticians, notably Richard Gill (who has a lot of interesting work on `quantum statistics'), I had never seen a paper from the second strand in a Statistics journal (only in Theoretical Physics journals). But there's a first time for everything...

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