You seem to have a problem with very small probabilities but not with very large numbers. I’ve also noticed this in Scott Alexander and others. If very small probabilities are zeros, then very large numbers are infinities.
You literally can not believe in an theory of physics that allows large amounts of computing power. If we discover that an existing theory like quantum physics allows us to create large computers, we will be forced to abandon it.
Sure. But since we know no such theory, there is no a priori reason to assume it exists with non-negligible probability.
Something like Solomonoff induction should generate perfectly sensible predictions about the world.
Nope, it doesn’t. If you apply Solomonoff induction to predict arbitrary integers, you get undefined expectations.
Solomonoff induction combined with an unbounded utility function gives undefined expectations. But Solomonoff induction combined with a bounded utility function can give defined expectations.
And Solomonoff induction by itself gives defined predictions.
Solomonoff induction combined with an unbounded utility function gives undefined expectations. But Solomonoff induction combined with a bounded utility function can give defined expectations.
Yes.
And Solomonoff induction by itself gives defined predictions.
If you try to use it to estimate the expectation of any unbounded variable, you get an undefined value.
Yes I understand that 3^^^3 is finite. But it’s so unfathomably large, it might as well be infinity to us mere mortals. To say an event has 1/3^^^3 is to say you are certain it will never happen, ever. No matter how much evidence you are provided. Even if the sky opens up and the voice of god bellows to you and says “ya its true”. Even if he comes down and explains why it is true to you, and shows you all the evidence you can imagine.
The word “negligible” is obscuring your true meaning. There is a massive—no, unfathomable—difference between 1/3^^^3 and “small” numbers like 1/10^80 (1 divided by the number of atoms in the universe.)
To use this method is to say there are hypotheses with relatively short descriptions which you will refuse to believe. Not just about muggers, but even simple things like theories of physics which might allow large amounts of computing power. Using this method, you might be forced to believe vastly more complicated and arbitrary theories that fit the data worse.
If you apply Solomonoff induction to predict arbitrary integers, you get undefined expectations.
Solomonoff inductions predictions will be perfectly reasonable, and I would trust them far more than any other method you can come up with. What you choose to do with the predictions could generate nonsense results. But that’s not a flaw with SI, but with your method.
You seem to have a problem with very small probabilities but not with very large numbers. I’ve also noticed this in Scott Alexander and others. If very small probabilities are zeros, then very large numbers are infinities.
Sure. But since we know no such theory, there is no a priori reason to assume it exists with non-negligible probability.
Nope, it doesn’t. If you apply Solomonoff induction to predict arbitrary integers, you get undefined expectations.
Solomonoff induction combined with an unbounded utility function gives undefined expectations. But Solomonoff induction combined with a bounded utility function can give defined expectations.
And Solomonoff induction by itself gives defined predictions.
Yes.
If you try to use it to estimate the expectation of any unbounded variable, you get an undefined value.
Probability is a bounded variable.
Yes, but I’m not sure I understand this comment. Why would you want to compute an expectation of a probability?
Yes I understand that 3^^^3 is finite. But it’s so unfathomably large, it might as well be infinity to us mere mortals. To say an event has 1/3^^^3 is to say you are certain it will never happen, ever. No matter how much evidence you are provided. Even if the sky opens up and the voice of god bellows to you and says “ya its true”. Even if he comes down and explains why it is true to you, and shows you all the evidence you can imagine.
The word “negligible” is obscuring your true meaning. There is a massive—no, unfathomable—difference between 1/3^^^3 and “small” numbers like 1/10^80 (1 divided by the number of atoms in the universe.)
To use this method is to say there are hypotheses with relatively short descriptions which you will refuse to believe. Not just about muggers, but even simple things like theories of physics which might allow large amounts of computing power. Using this method, you might be forced to believe vastly more complicated and arbitrary theories that fit the data worse.
Solomonoff inductions predictions will be perfectly reasonable, and I would trust them far more than any other method you can come up with. What you choose to do with the predictions could generate nonsense results. But that’s not a flaw with SI, but with your method.