As it turns out, there is an exact recipe for finding truth. It was discovered in the 1960s.
At our current state of knowledge, we cannot have any mathematical theorem about Solomonoff induction finding “truth” in our world, because we don’t know the mathematical underpinnings of our world. We can only have theorems about SI outperforming certain other methods of inference on all possible inputs.
The simplest of such theorems says SI outperforms all computable algorithms at the log-score game. This is not a very difficult feat because SI is allowed to be uncomputable. (SI can be constructed as a “mixture” of all computable algorithms, and arbitrarily changing the weights in the “mixture” will give you different but equally powerful versions of SI.)
A more interesting theorem says that SI is also optimal in its own class: possibly uncomputable distributions that can be specified as “lower-semicomputable semimeasures”. That’s genuinely surprising because most classes of distributions don’t have an optimal member within the class itself (see p.11 of Hutter’s slides, with only one “Yes” on the main diagonal). But it doesn’t seem very relevant to practice and somewhat undermines the special status of SI, because you can make universal mixtures of an even wider and more uncomputable class that will outperform SI the same way SI outperforms computable algorithms.
Thinking further along these lines, finding truth is likely to be an instrumental value rather than a terminal one, so we probably need to look at games other than the log-score game and see if SI can still make money when the rules for rewards and punishments are changed. This road leads to interesting unsolved questions, I wroteabout some of them :-)
Right now I’m not sure there’s any “deep truth” about applying SI to our world. In particular, there’s the troubling possibility that SI’s answers to your questions will depend on how you intend to use these answers!
When you describe a problem as conclusively solved when it’s not, you risk making your readers lose their sense of wonder. Maybe you could take a page from Feynman? He wasn’t afraid of giving the reader a glimpse of unsolved problems now and then. Eliezer also wrote about the perils of presenting a topic to students as if it were set in stone.
Right now I’m not sure there’s any “deep truth” about applying SI to our world.
Solomonoff induction isn’t particularly optimal at real tasks. Its performance on real-world tasks will depend somewhat on the choice of reference machine.
In particular, there’s the troubling possibility that SI’s answers to your questions will depend on how you intend to use these answers!
I don’t think so. I put my criticism of that idea there.
At our current state of knowledge, we cannot have any mathematical theorem about Solomonoff induction finding “truth” in our world, because we don’t know the mathematical underpinnings of our world. We can only have theorems about SI outperforming certain other methods of inference on all possible inputs.
The simplest of such theorems says SI outperforms all computable algorithms at the log-score game. This is not a very difficult feat because SI is allowed to be uncomputable. (SI can be constructed as a “mixture” of all computable algorithms, and arbitrarily changing the weights in the “mixture” will give you different but equally powerful versions of SI.)
A more interesting theorem says that SI is also optimal in its own class: possibly uncomputable distributions that can be specified as “lower-semicomputable semimeasures”. That’s genuinely surprising because most classes of distributions don’t have an optimal member within the class itself (see p.11 of Hutter’s slides, with only one “Yes” on the main diagonal). But it doesn’t seem very relevant to practice and somewhat undermines the special status of SI, because you can make universal mixtures of an even wider and more uncomputable class that will outperform SI the same way SI outperforms computable algorithms.
Thinking further along these lines, finding truth is likely to be an instrumental value rather than a terminal one, so we probably need to look at games other than the log-score game and see if SI can still make money when the rules for rewards and punishments are changed. This road leads to interesting unsolved questions, I wrote about some of them :-)
Right now I’m not sure there’s any “deep truth” about applying SI to our world. In particular, there’s the troubling possibility that SI’s answers to your questions will depend on how you intend to use these answers!
When you describe a problem as conclusively solved when it’s not, you risk making your readers lose their sense of wonder. Maybe you could take a page from Feynman? He wasn’t afraid of giving the reader a glimpse of unsolved problems now and then. Eliezer also wrote about the perils of presenting a topic to students as if it were set in stone.
Solomonoff induction isn’t particularly optimal at real tasks. Its performance on real-world tasks will depend somewhat on the choice of reference machine.
I don’t think so. I put my criticism of that idea there.