I just read a paper by Gerhard Schurz proposing an interesting resolution to the problem of induction. Download a PDF here.

Here's the abstract:

This article suggests a ‘best alternative’ justification of induction (in the sense of Reichenbach) which is based on meta‐induction. The meta‐inductivist applies the principle of induction to all competing prediction methods which are accessible to her. It is demonstrated, and illustrated by computer simulations, that there exist meta‐inductivistic prediction strategies whose success is approximately optimal among all accessible prediction methods in arbitrary possible worlds, and which dominate the success of every noninductive prediction strategy. The proposed justification of meta‐induction is mathematically analytical. It implies, however, an a posteriori justification of object‐induction based on the experiences in our world.

Here's Schurz's description of meta-inductivism:

The meta‐inductivist (MI) applies the inductive method at the level of competing prediction methods. More precisely, the meta‐inductivist bases her predictions on the predictions and the observed success rates of the other (non‐MI) players and tries to derive therefrom an ‘optimal’ prediction. The simplest type of MI predicts what the presently best prediction method predicts, but one can construct much more refined kinds of meta‐inductivistic prediction strategies.

One should expect that for meta‐induction the chances of demonstrating optimality are much better than for object‐induction. The crucial question of this article will be: is it possible to design a version of meta‐induction which can be proved to be an (approximately) optimal prediction method? The significance of this question for the problem of induction is this: if the answer is positive, then at least meta‐induction would have a rational and noncircular justification based on a mathematical‐analytic argument. But this analytic justification of meta‐induction would at the same time yield an a posteriori justification of object‐induction in the real world: for we know by experience that in the real world, noninductive prediction strategies have not been successful so far, hence it would be meta‐inductively justified to favor object‐inductivistic strategies.

Here's the conclusion:

While one‐favorite meta‐inductive strategies are optimal only under certain restrictions, weighted‐average meta‐induction has turned out to be universally optimal...

In conclusion, I think the achieved optimality results on meta‐induction are strong enough to show that a noncircular justification of (meta‐)induction can be successful. This justification does not show that meta‐induction must be successful (in a strict or probabilistic sense), but it favors the meta‐inductivistic strategy against all other accessible competitors. This is sufficient for justificational purposes, without being in dissent with any of Hume’s skeptical arguments. The given justification of meta‐induction is mathematically‐analytic (or ‘a priori’), insofar it does not make any assumptions about the nature of the considered worlds except from practically evident assumptions about prediction games, such that its players can perform calculations, can observe past events, and are free to decide. However, as we have explained in Section 2, this analytic justification of meta‐induction implies an a posteriori justification of object‐induction in our real word, because so far object‐induction has turned out to be the most successful prediction strategy. This argument is no longer circular, given that we have a noncircular justification of meta‐induction—and we have it.

The major advantage of the meta‐inductivistic approach is its radical openness towards all kinds of possibilities. In my view, this radical openness is a sign of all good foundation‐oriented (instead of ‘foundationalistic’) programs in epistemology. Unlike in Rescher’s “initial justification” of induction (1980, 82), meta‐induction does not exclude esoteric world‐views or prediction methods from the start. Such an a priori exclusion would prevent a constructive dialog between a scientific philosopher and an esoteric‐minded person. Meta‐induction takes all these possible world‐views initially seriously and argues: wherever the ‘ultimate truth’ lies, you should in any case employ meta‐induction because it is universally optimal among all accessible prediction methods.

Many readers will still uphold skeptical reservations. They will ask: how can it ever be possible to prove that a strategy is optimal with respect to every other accessible strategy in every possible world—without assuming anything about the nature of alternative strategies and possible worlds? My heuristic answer to this skeptical challenge is as follows: this is possible for meta‐inductive strategies because these strategies are universal learners: whenever they are confronted with a so far better strategy, they can imitate the better strategy (output‐accessibility) or even learn to reproduce it (internal accessibility).