p = 0 and p = 1, with a maximum somewhere inbetween, say at p = 1/e.

Blasphemy ! -p*log(p) - (1-p)*log(1-p) goes maximal at at p=1/2, the point of 'least information', or equal probability for p=(1-p)=1/2. Wouter. -----Original Message----- From: Marc LeBrun [mailto:mlb@fxpt.com] Sent: vrijdag 7 maart 2003 18:24 To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Why is e the "best" base? + A symmetrical way to treat digits 0,1,2 base e

=mcintosh@servidor.unam.mx Entropy is p ln(p) summed over alternatives. It vanishes for p = 0 and p = 1, with a maximum somewhere inbetween, say at p = 1/e. An interesting thing about this formula is that the maximum is independent of the base used for the logarithm.

However, are you sure about that p factor? I thought the information content of a message was, roughly, how "surprising" it was, hence simply -ln(p) (which is why information gets called "negative entropy"). Getting an impossible message would be a miraculous epiphany, containing infinite information (alas of the form "everything you know is wrong!"<;-) I'll try to dig out my copy of Shannon's original paper and see what he said. By the way, is there an analogous "quantum entropy", which would involve taking the log of the "complex probability"? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.