Speed Prior-Based Inductive Inference
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## Speed Prior-Based Inductive Inference

Given S, as we observe an initial segment of some string, which is the most likely continuation? Consider x's finite continuations . According to Bayes (compare Equation (15)),

 (48)

where is the measure of z2, given z1. Having observed x we will predict those y that maximize . Which are those? In what follows, we will confirm the intuition that for the only probable continuations of xn are those with fast programs. The sheer number of slowly'' computable strings cannot balance the speed advantage of more quickly'' computable strings with equal beginnings.

Definition 6.4 (tex2html_wrap_inline$p < k_i x$ etc.)   Write if finite program p () computes x within less than k steps, and if it does so within PHASE i of FAST. Similarly for and (at most k steps), , (exactly k steps), , (at least k steps), (more than k steps).

Theorem 6.1   Suppose is S-describable, and outputs xn within at most f(n) steps for all n, and g(n) > O(f(n)). Then

Proof. Since no program that requires at least g(n) steps for producing xn can compute xn in a phase with number < log g(n), we have

Here we have used the Kraft inequality [#!Kraft:49!#] to obtain a rough upper bound for the enumerator: when no p is prefix of another one, then

Hence, if we know a rather fast finite program px for x, then Theorem 6.1 allows for predicting: if we observe some xn (n sufficiently large) then it is very unlikely that it was produced by an x-computing algorithm much slower than px.

Among the fastest algorithms for x is FAST itself, which is at least as fast as px, save for a constant factor. It outputs xn after O(2Kt(xn)) steps. Therefore Theorem 6.1 tells us:

Corollary 6.1   Let be S-describable. For , with probability 1 the continuation of xn is computable within O(2Kt(xn)) steps.

Given observation x with , we predict a continuation y with minimal Kt(xy).

Example 6.1   Consider Example 1.2 and Equation (1). According to the weak anthropic principle, the conditional probability of a particular observer finding herself in one of the universes compatible with her existence equals 1. Given S, we predict a universe with minimal Kt. Short futures are more likely than long ones: the probability that the universe's history so far will extend beyond the one computable in the current phase of FAST (that is, it will be prolongated into the next phase) is at most 50 %. Infinite futures have measure zero.

Next: Practical Applications of Algorithm Up: Temporal Complexity Previous: Speed Prior S and
Juergen Schmidhuber
2001-01-09

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