# Re: Predictions & duplications

```
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/

```