> From: Juho Pennanen <[EMAIL PROTECTED]> > So there may be no 'uniform probability distribution' on the set of all > strings, but there is the natural probability measure, that is in many > cases exactly as useful.
Sure, I agree, measures are useful; I'm using them all the time. But in general they are _not_ the same thing as probability distributions. > From: Russell Standish <[EMAIL PROTECTED]> > The only reason for not accepting the simplest thing is if it can be > shown to be logically inconsistent. This far, you have shown no such > thing, but rather demonstrated an enormous confusion between measure > and probability distribution. Russell, this is really too much; please read any intro to measure theory, e.g., the one by Halmos. Measures in general are _not_ probability distributions. They are more than that; you can view them as _sets_ of probability distributions on sets that are related to each other in a specific way. Probability density functions define measures and therefore in general aren't probability distributions either. The uniform PDF on [0..1] yields the perfect trivial example; that's exactly why I used it before. In the computability context, compare LI/Vitanyi 1997, chapters 4.3 (discrete sample space with (semi)distribution m(x)) vs chapter 4.5 (continuous sample space with (semi)measure M(x)). > 1) Halting theorem implies the set of halting programs is of measure > zero within the set of all programs You mean, given a uniform measure? But why should you need the halting theorem for this trivial statement? In fact, what exactly do you mean by halting theorem? (Schoenfield's book provides a good intro to mathematical logic and computability theory.) > The GP program (aka Universal Dovetailer) is not the simplest > thing. The simplest describable thing is the set of all descriptions, > that simply exist without being generated. That has precisely zero > information or complexity, whereas the UD has some complexity of the > order of a few 100 bits. The simplest prior is the uniform one, ie > there is no bias whatsoever in the selection. Exist without being generated? Precisely zero information or complexity? But with respect to what? Any traditional definition of information/simplicity requires the specification of a formal description method, such as a Turing machine. You don't need one? Then how do you define information or complexity? How exactly do you define "simple" ?
