# Re: how to do something with really small probability?

From: Wei Dai (weidai@weidai.com)
Date: Mon Nov 05 2007 - 17:06:29 MST

Rolf Nelson wrote:
> How do you know *anything*? You have a Bayesian "prior distribution",
> which may include anthropic reasoning.

I'm assuming Bayesian reasoning as well. I don't think anthropic reasoning
is relevant to this issue.

> Obviously if a bounded-rationality agent is aware that it's a
> uniformly random program, then once it has seen that x is its output,
> it should (if it is sophisticated, and it has nothing better to do
> with its time) give x a probability on the order of the amount of
> Chaitin's Omega that it doesn't know. So what? You're begging the
> question of why it had this prior in the first place. The prior
> certainly isn't true of the programs running on my PC; none of my
> programs are drawn from uniformly random distributions (not even
> Microsoft Word).

I was using the standard prior for Solomonoff Induction. I see Nick Hay has
written an exposition of the concept at
http://www.intelligence.org/blog/2007/06/25/solomonoff-induction/. Quoting from
it:

Solomonoff induction predicts sequences by assuming they are produced by a
random program. The program is generated by selecting each character
randomly until we reach the end of the program.

>> Now is it
>> possible that SI can take an arbitrary string x and tell us whether P(x)
>> <
>> 1/2^(2^100)?
>
> Underspecified. If by "probability" you only mean "something that
> obeys the Probability Axioms, and is also sometimes useful", then
> sure. If an agent has bounded rationality, it can consistently say
> "there is a 1/2 probability that any number between 1 and 10 is an
> even number. There is a 1/2 probability that 5 is an even number.
> There is a 1/2 probability that 6 is an even number."

This P is supposed to be the same function as before (i.e., the standard
prior for Solomonoff Induction).

Does that clear up the point I was trying to make?

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