This shows that probabilities which are apparently very small can rapidly shift to being quite large with the influx of new information.
Isn’t there a simpler way of making this point, without going into the properties of pi and the history of its computation? For example, I could write a small program that will fill my screen with a random n digit numbers at time X. Before time X, the probability of me seeing any particular number is 10^-n, and after time X it’s 1 for one number and 0 for others.
But I’m having trouble seeing the practical significance of this. It would be interesting to see Anna’s side of this conversation, to get a better idea of what you guys were talking about...
Isn’t there a simpler way of making this point, without going into the properties of pi and the history of its computation? For example, I could write a small program that will fill my screen with a random n digit numbers at time X. Before time X, the probability of me seeing any particular number is 10^-n, and after time X it’s 1 for one number and 0 for others.
It may be that my exposition could be improved. I’m not sure whether or not the use of a particular real number is central to the line of thinking that I’m interested in exploring.
If any two independent people compute the first thousand digits of pi then the results should be identical whereas if two people generate 1000 random digits then their results will not be identical. In this way “the first 1000 digits of pi” is well defined in a way that “1000 random digits” is not.
Note that it’s possible to think that a string of n digits is probably the first n digits of pi and then learn that there was an error in the coding/algorithm used which causes a dramatic drop in the probability that the string of digits is correct whereas it’s not possible to learn that a string of random digits is “wrong.”One may learn that the algorithm used was not a good random number generator but that’s not quite the same thing...
Some readers may find the properties of pi and the history of its computation to be of independent interest.
But I’m having trouble seeing the practical significance of this. It would be interesting to see Anna’s side of this conversation, to get a better idea of what you guys were talking about...
It was just a casual conversation; a tangent to an unrelated subject. But my interest in the topic is that I think that reasoning about small probabilities may be important to x-risk reduction and I want to get a feel for the types of errors & the times of valid heuristics attached to the domain with a view toward thinking about x-risk reduction.
Isn’t there a simpler way of making this point, without going into the properties of pi and the history of its computation? For example, I could write a small program that will fill my screen with a random n digit numbers at time X. Before time X, the probability of me seeing any particular number is 10^-n, and after time X it’s 1 for one number and 0 for others.
But I’m having trouble seeing the practical significance of this. It would be interesting to see Anna’s side of this conversation, to get a better idea of what you guys were talking about...
It may be that my exposition could be improved. I’m not sure whether or not the use of a particular real number is central to the line of thinking that I’m interested in exploring.
If any two independent people compute the first thousand digits of pi then the results should be identical whereas if two people generate 1000 random digits then their results will not be identical. In this way “the first 1000 digits of pi” is well defined in a way that “1000 random digits” is not.
Note that it’s possible to think that a string of n digits is probably the first n digits of pi and then learn that there was an error in the coding/algorithm used which causes a dramatic drop in the probability that the string of digits is correct whereas it’s not possible to learn that a string of random digits is “wrong.”One may learn that the algorithm used was not a good random number generator but that’s not quite the same thing...
Some readers may find the properties of pi and the history of its computation to be of independent interest.
It was just a casual conversation; a tangent to an unrelated subject. But my interest in the topic is that I think that reasoning about small probabilities may be important to x-risk reduction and I want to get a feel for the types of errors & the times of valid heuristics attached to the domain with a view toward thinking about x-risk reduction.