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From: <juergen.domain.name.hidden>

Date: Tue, 16 Oct 2001 17:40:03 +0200

Confusion about what's a measure?

What's a distribution?

Simple but important!

For bitstrings x:

M measure:

M(empty string)=1

M(x) = M(x0)+M(x1) nonnegative for all finite x.

P probability distribution:

Sum_x P(x) = 1; P(x) nonnegative

Date: Tue, 16 Oct 2001 17:40:03 +0200

Confusion about what's a measure?

What's a distribution?

Simple but important!

For bitstrings x:

M measure:

M(empty string)=1

M(x) = M(x0)+M(x1) nonnegative for all finite x.

P probability distribution:

Sum_x P(x) = 1; P(x) nonnegative

--- M semimeasure - replace "=" by ">=": M(x) >= M(x0)+M(x1) P semidistribution - replace: Sum_x P(x) <= 1 --- Examples: 1. Distribution: E.g., integers n: P(n) = 6/(Pi n^2) 2. Semidistribution: m(x) = probability of guessing a halting program for x (BTW, Hal, this was first published by Levin in 1974, not by Chaitin in 1975) 3. Measure: E.g., each x of size n gets weight 2^-n 4. Semimeasure: E.g., mu^M(x) = probability of guessing a halting or nonhalting monotone TM program whose output starts with x Check out: Measures and Probability Distributions Section 4 of "Algorithmic TOEs" http://www.idsia.ch/~juergen/toesv2/node15.html Juergen Schmidhuber http://www.idsia.ch/~juergen/ http://www.idsia.ch/~juergen/everything/html.html http://www.idsia.ch/~juergen/toesv2/Received on Tue Oct 16 2001 - 08:49:08 PDT

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