Open Thread, May 1-14, 2013

If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

Hi, my name is Jason, this is my first post. I have recently been reading about 2 subjects here, Calibration and Solomoff Induction; reading them together has given me the following question:

How well-calibrated would Solomonoff Induction be if it could actually be calculated?

That is to say, if one generated priors on a whole bunch of questions based on information complexity measured in bits - if you took all the hypotheses that were measured at 10% likely - would 10% of those actually turn out to be correct?

I don't immediately see why Solomonoff Induction should be expected to be well-calibrated. It appears to just be a formalization of Occam's Razor, which itself is just a rule of thumb. But if it turned out not to be well-calibrated, it would not be a very good "recipe for truth." What am I missing?

Viliam_Bur makes a great run-down of what's going on. For a more detailed introduction though, see this post explaining Solomonoff Induction, or perhaps you'd prefer to jump straight to this paragraph (Solomonoff's Lightsaber) that contains an explanation of why shorter (simpler) hypotheses are more likely under Solomonoff Induction.

To make the bridge between that and what Viliam is saying, basically, if we consider all mathematically possible universes, then half the universes will start with a 1, and the other half will start with a 0. Then a quarter wi... (read more)

6Viliam_Bur13y

Solomonoff Induction could be well-calibrated across mathematically possible universes. If a hypothesis has a probability 10%, you should expect it to be true in 10% of the universes. Important thing is that Solomonoff priors are just a starting point in our reasoning. Then we update on evidence, which is at least as important as having reasonable priors. If it does not seem well calibrated, that is because you can't get good calibration without using evidence. Imagine that at this moment you are teleported to another universe with completely different laws of physics... do you expect any other method to work better than Solomonoff Induction? Yes, gradually you get data about the new universe and improve your model. But that's exactly what you are supposed to do with Solomonoff priors. You wouldn't predictable get better results by starting from different priors. [...] To me it seems that Occam's Razor is a rule of thumb, and Solomonoff Induction is a mathematical background explaining why the rule of thumb works. (OR: "Choose the most simple hypothesis that fits your data." Me: "Okay, but why?" SI: "Because it is more likely to be the correct one.") [...] You can't get a good "recipe for truth" without actually looking at the evidence. Solomonoff Induction is the best thing you can do without the evidence (or before you start taking the evidence into account). Essentially, the Solomonoff Induction will help you avoid the following problems: * Getting inconsistent results. For example, if you instead supposed that "if I don't have any data confirming or rejecting a hypothesis, I will always assume its prior probability is 50%", then if I give you two new hypotheses X and Y without any data, you are supposed to think that p(X) = 0.5 and p(Y) = 0.5, but also e.g. p(X and Y) = 0.5 (because "X and Y" is also a hypothesis you don't have any data about). * Giving so extremely low probability to a reasonable hypothesis that available evidence cannot convince you