So, I've been hearing a lot about the awesomeness of Solomonoff induction, at least as a theoretical framework. However, my admittedly limited understanding of Solomonoff induction suggests that it would form an epicly bad hypothesis if given a random string. So my question is, if I misunderstood, how does it deal with randomness? And if I understood correctly, isn't this a rather large gap?
Edit: Thanks for all the comments! My new understanding is that Solomonoff induction is able to understand that it is dealing with a random string (because it finds itself giving equal weight to programs that output a 1 or a 0 for the next bit), but despite that is designed to keep looking for a pattern forever. While this is a horrible use of resources, the SI is a theoretical framework that has infinite resources, so that's a meaningless criticism. Overall this seems acceptable, though if you want to actually implement a SI you'll need to instruct it on giving up. Furthermore, the SI will not include randomness as a distinct function in its hypothesis, which could lead to improper weighting of priors, but will still have good predictive power -- and considering that Solomonoff induction was only meant for computable functions, this is a pretty good result.
But it seems to me rather different to assume you can do any finite amount of calculation, vs relying on things that can only be done with infinite calculation. Can we ever have a hope of having infinite resources?
I think epsilon.
Just to clarify, though: in using universal induction, every hypothesis is finite in size, thereby at no point some process needs to run an infinite program to discover its output. The infinite part is of course the size of memory space, used to hold an infinite prior, and the infinite speed of its update.