solomonoff-induction

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Solomonoff induction is an inference system defined by Ray Solomonoff that will learn to correctly predict any computable sequence with only the absolute minimum amount of data. This system, in a certain sense, is the perfect universal prediction algorithm. 

To summarize it very informally, Solomonoff induction works by:

Weighting hypotheses by simplicity, the system automatically incorporates a form of Occam's razor [? · GW], which is why it has been playfully referred to as Solomonoff's lightsaber.

Solomonoff induction gets off the ground with a solution to the "problem of the priors". Suppose that you stand before a universal prefix Turing machine U. You are interested in a certain finite output string y0. In particular, you want to know the probability that U will produce the output y0 given a random input tape. This probability is the Solomonoff a priori probability of y0.

More precisely, suppose that a particular infinite input string x0 is about to be fed into U. However, you know nothing about x0 other than that each term of the string is either 0 or 1. As far as your state of knowledge is concerned, the ith digit of x0 is as likely to be 0 as it is to be 1, for all i = 1, 2, …. You want to find the a priori probability m(y0) of the following proposition:

(*) If U takes in x0 as input, then U will produce output y0 and then halt.

Unfortunately, computing the exact value of m(y0) would require solving the halting problem, which is undecidable. Nonetheless, it is easy to derive an expression for m(y0). If U halts on an infinite input string x, then U must read only a finite initial segment of x, after which U immediately halts. We call a finite string p a self-delimiting program if and only if there exists an infinite input string x beginning with p such that U halts on x immediately after reading to the end of p. The set 𝒫 of self-delimiting programs is the prefix code for U. It is the determination of the elements of 𝒫 that requires a solution to the halting problem.

Given p ∈ 𝒫, we write "prog (x0) = p" to express the proposition that x0 begins with p, and we write "U(p) = y0" to express the proposition that U produces output y0, and then halts, when fed any input beginning with p. Proposition (*) is then equivalent to the exclusive disjunction


⋁p ∈ 𝒫: U(p) = y0(prog (x0) = p).
Since x0 was chosen at random from {0, 1}ω, we take the probability of prog (x0) = p to be 2 − ℓ(p), where ℓ(p) is the length of p as a bit string. Hence, the probability of (*) is


m(y0) := ∑p ∈ 𝒫: U(p) = y02 − ℓ(p).
 

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References