Prediction Using a Universal Algorithmic Prior Based on the Shortest Way of Describing Objects

Roughly fourty years ago Solomonoff started the theory of
universal optimal induction based on the apparently harmless
simplicity assumption that is computable [62].
While Equation (1) makes predictions of
the entire future, given the past,
Solomonoff [63] focuses just on the next
bit in a sequence. Although this provokes surprisingly nontrivial
problems associated with translating the bitwise approach to alphabets
other than the binary one -- this was achieved only recently
[20] -- it is sufficient for obtaining essential
insights. Given an observed bitstring , Solomonoff assumes the data are drawn
according to a recursive measure ; that is, there is a program for
a universal Turing machine that reads and
computes and halts. He estimates the probability of the next bit
(assuming there will be one), using the remarkable, well-studied,
enumerable prior
[62,77,63,15,31]

(3) |

**Recent Loss Bounds for Universal Prediction.**
A more general recent result is this. Assume we do know that
is in some set of distributions. Choose a fixed weight
for each in such that the add up to 1 (for simplicity,
let be countable). Then construct the Bayesmix
, and predict using instead of the
optimal but unknown .
How wrong is it to do that? The recent work of Hutter
provides general and sharp (!) loss bounds [21]:

Let and be the total expected unit losses of the -predictor and the p-predictor, respectively, for the first events. Then is at most of the order of . That is, is not much worse than . And in general, no other predictor can do better than that! In particular, if is deterministic, then the -predictor soon won't make any errors any more.

If contains *all* recursively computable distributions, then
becomes the celebrated enumerable universal prior. That is, after
decades of somewhat stagnating research we now have sharp loss
bounds for Solomonoff's universal induction
scheme (compare recent work of Merhav and Feder [33]).

Solomonoff's approach, however, is uncomputable. To obtain a feasible approach, reduce M to what you get if you, say, just add up weighted estimated future finance data probabilities generated by 1000 commercial stock-market prediction software packages. If only one of the probability distributions happens to be close to the true one (but you do not know which) you still should get rich.

Note that the approach is much more general than what is normally done in traditional statistical learning theory, e.g., [69], where the often quite unrealistic assumption is that the observations are statistically independent.