Solomonoff induction

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Solomonoff induction

Ray Solomonoff defined an inference system 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:

Starting with all possible hypotheses (sequences) as represented by computer programs (that generate those sequences), weighted by their simplicity (2^-n, where n is the program length);
Discarding those hypotheses that are inconsistent with the data.

Weighting hypotheses by simplicity, the system automatically incorporates a form of Occam's razor, 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 $\text{[math]}$ . You are interested in a certain finite output string $\text{[math]}$ . In particular, you want to know the probability that $\text{[math]}$ will produce the output $\text{[math]}$ given a random input tape. This probability is the Solomonoff a priori probability of $\text{[math]}$ .

More precisely, suppose that a particular infinite input string $\text{[math]}$ is about to be fed into $\text{[math]}$ . However, you know nothing about $\text{[math]}$ other than that each term of the string is either $\text{[math]}$ or $\text{[math]}$ . As far as your state of knowledge is concerned, the $\text{[math]}$ th digit of $\text{[math]}$ is as likely to be $\text{[math]}$ as it is to be $\text{[math]}$ , for all $\text{[math]}$ . You want to find the a priori probability $\text{[math]}$ of the following proposition:

(*) If $\text{[math]}$ takes in $\text{[math]}$ as input, then $\text{[math]}$ will produce output $\text{[math]}$ and then halt.

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

Given $\text{[math]}$ , we write " $\text{[math]}$ " to express the proposition that $\text{[math]}$ begins with $\text{[math]}$ , and we write " $\text{[math]}$ " to express the proposition that $\text{[math]}$ produces output $\text{[math]}$ , and then halts, when fed any input beginning with $\text{[math]}$ . Proposition (*) is then equivalent to the exclusive disjunction

\text{[math]}

Since $\text{[math]}$ was chosen at random from $\text{[math]}$ , we take the probability of $\text{[math]}$ to be $\text{[math]}$ , where $\text{[math]}$ is the length of $\text{[math]}$ as a bit string. Hence, the probability of (*) is