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Measures and Probability Distributions

Suppose x represents the history of our universe up until now. What is its most likely continuation $y \in B^{\sharp}$? Bayes' theorem yields

 \begin{displaymath}P(xy \mid x)
= \frac{P(x \mid xy) P(xy)} {\sum_{z \in B^{\sharp}} P(xz)}
= \frac{P(xy)} {N(x)}
\propto P(xy)
\end{displaymath} (15)

where $P(z^2 \mid z^1)$ is the probability of z2, given knowledge of z1, and

\begin{displaymath}N(x) = \sum_{z \in B^{\sharp}} P(xz)
\end{displaymath} (16)

is a normalizing factor. The most likely continuation y is determined by P(xy), the prior probability of xy -- compare the similar Equation (1). Now what are the formally describable ways of assigning prior probabilities to universes? In what follows we will first consider describable semimeasures on B*, then probability distributions on $B^{\sharp}$.


Juergen Schmidhuber

Related links: In the beginning was the code! - Zuse's thesis - Life, the universe, and everything - Generalized Algorithmic Information - Speed Prior - The New AI