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Kindly comments on Open Problems Related to Solomonoff Induction - Less Wrong
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Comments (102)
The hypotheses are encoded in a prefix-free way: no hypothesis is a prefix of another hypothesis.
In particular, this eliminates your example: if both strings of length 1 ("0" and "1") are valid hypotheses, then no string of longer length can be a valid hypothesis (then we can take Omega=1 and assign a probability of 1/2 to each). Or we could have "0", "10", and "110" be valid hypotheses; then we can take Omega = 7/8 and assign probabilities of 4/7, 2/7 and 1/7 to these.
In general, the prefix-free condition ensures that the sum over all hypotheses converges, by Kraft's inequality, which is the real heavy lifter here; the constant is merely there to make the sum over all hypotheses converge to 1 instead.
You might imagine that hypotheses are required to be grammatically correct English sentences (I guess encoded in ASCII or something). In that case, no hypothesis is a prefix of another, because each hypothesis ends with a period.
Right; I forgot that it used a prefix-free encoding. Apologies if the answer to this is painfully obvious, but does having a prefix-free encoding entail that there is a finite number of possible hypotheses?
It doesn't: for example, the set of strings 0, 10, 110, 1110, 11110, 111110, ... is infinite and prefix-free. So is the set of C-style (null-terminated) strings.
I see. Does the method of normalization you gave work even when there is an infinite number of hypotheses?
It does. The infinite sum will always converge, so by normalizing we can make it converge to 1.
Take gjm's approach to explaining the normalization, in which the initial weight of 2^-length assigned to a hypothesis is the probability that you obtain that hypothesis by choosing bits at random. Then the normalization is just a conditional probability: you divide by the probability that, when choosing bits at random, you do eventually end up hitting a hypothesis.
Remark: these are special cases of the schema "all strings that contain exactly one instance of pattern X in possible positions Y", where in the first case X is "0" and Y is "anywhere" and in the second X is "00000000" and Y is "any position that's a multiple of 8 bits".
(Of course there are plenty of other prefix-free sets of bitstrings. For instance: interpret blocks of 8 bits as characters as in Kindly's second example, and say that a valid program is anything that begins with a "(", has properly matched parentheses, and ends with the ")" matching the first "(". This doesn't have the convenient property that every bitstring begins with a valid program; for examples that do, take the exponential Golomb coding or the Elias omega coding of natural numbers.