The utility of logarithms? I've always thought using a slide rule was more esthetic than pushing buttons on a calculator. Perhaps few people still know what a slide rule can do. Of course, the result might be a little less accurate . . .
Best, Joseph -----Original Message----- From: Fis [mailto:fis-boun...@listas.unizar.es] On Behalf Of Mark Johnson Sent: dimanche, 3 juin 2018 22:07 To: Krassimir Markov Cc: Foundation of Information Science; Sungchul Ji Subject: Re: [Fis] No, this is not the reason. Dear Krassimir and Sungchul, I suppose this bears out Von Neumann's tongue-in-cheek advice to Shannon! (http://www.eoht.info/page/Neumann-Shannon+anecdote) Krassimir, just to ask about Boltzmann's use of the logs... I first understood this to be a measure of the probability distribution of a whole thermodynamic system which factorises into the product of probabilities of microstates in the system. Hence the logs (and hence Shannon equating of "microstate" for "alphabet", which seems reasonable at first glance)... EXCEPT I very much like the explanation that Bob Ulanowicz gives here (in http://www.mdpi.com/2078-2489/2/4/624) - which doesn't mention the factorising of the probabilities of microstates, but instead argues that -log (p(i)) gives a value for what isn't there (the "apophatic", "absence"), and Bob's criticism of Shannon for inverting this by turning his H into a measure of surprise: "Boltzmann described a system of rarefied, non-interacting particles in probabilistic fashion. Probability theory quantifies the degree to which state i is present by a measure, p(i). Conventionally, this value is normalized to fall between zero and one by dividing the number of times that i has occurred by the total number of observations. Under this "frequentist" convention, the probability of i not occurring becomes (1 - p(i)). Boltzmann's genius, however, was in abjuring this conventional measure of non-occurrence in favor of the negative of the logarithm of pi. (It should be noted that -log(p(i)) and (1 - p(i)) vary in uniform fashion, i.e., a one-to-one relationship between the two functions exists). His choice imposed a strong asymmetry upon matters. Conventionally, calculating the average nonbeing in the system using (1 - p(i)) results in the symmetrical parabolic function (p(i) - p(i)^2). If, however, one calculates average absence using Boltzmann's measure, the result becomes skewed towards smaller p(i) (or larger [1 - p(i)]), i.e., towards nonbeing." It's such a useful equation, and I agree, "Why are the logs there?" is an important question. Best wishes, Mark On 3 June 2018 at 20:22, Krassimir Markov <mar...@foibg.com> wrote: > Dear Sung, > > You wrote: >> I think the main reason that we express 'information' as a logarithmic > function of the number of choices available, n, may be because the human > brain finds it easier to remember (and communicate and reason with) 10 > than 10000000000, or 100 than 1000000000. . . . 00000, etc. >> > > No, this is not the reason. > The correct answer is that Shannon assume the n=0 as possible !!! > Because of this, to avoid dividing by zero he used log(s). > But this is impossible and many years the world works with log(s) not > understanding why ! > > log(s) is(are) no needed. > > It is more clear and easy to work without log(s) :=) > > Friendly greetings > Krassimir > > > > > _______________________________________________ > Fis mailing list > Fis@listas.unizar.es > http://listas.unizar.es/cgi-bin/mailman/listinfo/fis -- Dr. Mark William Johnson Institute of Learning and Teaching Faculty of Health and Life Sciences University of Liverpool Phone: 07786 064505 Email: johnsonm...@gmail.com Blog: http://dailyimprovisation.blogspot.com _______________________________________________ Fis mailing list Fis@listas.unizar.es http://listas.unizar.es/cgi-bin/mailman/listinfo/fis --- L'absence de virus dans ce courrier électronique a été vérifiée par le logiciel antivirus Avast. https://www.avast.com/antivirus _______________________________________________ Fis mailing list Fis@listas.unizar.es http://listas.unizar.es/cgi-bin/mailman/listinfo/fis