I'm confused. Assuming that I "believe in" the validity of what I have been told of quantum mechanics, I fully expect that a million quantum coin tosses will generate an incompressible string. Are you suggesting that I cannot simultaneously believe in the validity of QM and also believe in the efficacy of Solomonoff induction - when applied to data which is "best explained" as causally random?
Off the top of my head, I am inclined to agree with this suggestion, which in turn suggests that Si is flawed. We need a variant of Si which allows Douglas_Knight's simple fair coins, without thereby offering a simple explanation of everything. Or, we need to discard the whole Si concept as inappropriate in our non-deterministic universe.
The randomness of a source of information is not an empirical fact which we can discover and test - rather, it is an assumption that we impose upon our model of the data. It is a null hypothesis for which we cannot find Bayesian evidence - we can at best fail to reject it. (I hope the Popper-clippers don't hear me say that!). Maybe what our revised Si should be looking for is the simplest explanation for data D[0] thru D[n], which explanation is not refuted by data D[n+1] thru D[n+k].
ETA: Whoops. That suggestion doesn't work. The simplest such explanation will always be that everything is random.
Off the top of my head, I am inclined to agree with this suggestion, which in turn suggests that Si is flawed. We need a variant of Si which allows Douglas_Knight's simple fair coins, without thereby offering a simple explanation of everything. Or, we need to discard the whole Si concept as inappropriate in our non-deterministic universe.
I don't think the universe shows any signs of being non-deterministic. The laws of physics as we understand them (e.g. the wave equation) are deterministic. So, Solomonoff induction is not broken.
You're about to flip a quantum coin a million times (these days you can even do it on the internet). What's your estimate of the K-complexity of the resulting string, conditional on everything else you've observed in your life so far? The Born rule, combined with the usual counting argument, implies you should say "about 1 million". The universal prior implies you should say "substantially less than 1 million". Which will it be?
EDIT: Wei Dai's comment explains why this post is wrong.