Natural Versus "universal" Probability, Complexity, And Entropy

The interrelated concepts of information. entropy, probability, aiid universal coniyutaliou eiijoy w r y wide applicatioii. I t is wvrth iroting, tliutcgh, t l t d the successful applications and theorems are inostly restricted to systeiits lcaviirg a zwst riuinler of par*licles or syinbols. For smal ler sys tems, it i s not clear how i o quanliffy th.e infom~atioii./eiitropy/probabtlziy precisely. W e uiyue that these are trot jusl. acigue, (icadt nric qtre.si.iotia, b-ut svirset irines (befits it e p 1.0 hlcnrs a I & praciice. Malheiiraiics and computer science al1o.w u s lo coiistrucd nisy nrriinher of itiequitrnlend probability n t m sure.q. E v e s ‘90-called “unii~er.sal” prohability niwnsiires can be quite iiiequivaleiit. Deciding tuhicli of these best deacribea the red world ia a t m k for enrpirical . w i t w e . As illuatratioiss uie coiisider the (1) itijorniation coiit ea l of the genome and (,?) a practical “lcariiing front ezamplcs ” task. Additional kcywords: ITniversal Turing Afnch,ine, Uti iversnl Data (?oiicpressivri., Viiive w a l P i i o r, A 1goiiih.iiric Complexity, Iiolrnogorotr Complexity, Macliiiie Leariiiirg.