My motivations for asking this question are philosophical in nature. I'm by no means a computer scientist though, and I feel as though this question should be answered by someone who is since it's one thing to read about a subject second hand and another to understand it first hand. I'm an A-level student of philosophy, physics and mathematics (UK qualification) if that helps you formulate an answer at all.
The question relates to the problem of induction in that, we have little reason to believe the universe is uniform and apply that expectation to science and inference. However, I've seen the claim made that this guy Solomonoff created a theory of induction that uses a Bayesian framework as well as something called 'Kolomogrov complexity' as an objective measure of 'complexity' in a model or hypothesis. For example, if we observe over and over again that all A's are B's, we might be tempted to formulate the hypothesis:
"All A's are B's",
This would be afforded the maximum credence from a bayesian framework since it predicts that A's are B's with a probability of 1 and we're attaching ourselves to the likelihood lover's principle. However, you could also formulate the hypothesis:
"All A's are B's until a time t where they are C."
And there are infinitely many of these hypothesis since the 'C' can represent literally anything. This has a bayesian multiplier equal to the previous hypothesis so technically have an equal credence if we don't prefer one of these in our priors. And the only way (it would seem) to decide between the two is to see if hypothesis 2 shows it's prediction at time t. So we would otherwise have to always wait until time t to see if hypothesis 1 is correct.
My question can be split up into three parts:
1) Is "Kolmogrov complexity" generally considered a genuine formulation of an objective definition for complexity/simplicity?
2) Would the former hypothesis, with such an understanding of complexity be considered more 'simple'?
3) How would this understanding of complexity be used practically when distinguishing between hypotheses? Or is it just obvious that the former type of hypothesis is more simple?
Although this is really a philosophical question, I suppose I would also like to know if anyone here has any ideas about why we should prefer more simple hypotheses. And perhaps, how would you increase your bayesian prior probability given some measure of complexity for a hypothesis?
Thank you in advance for any answers.