Pick a random rational number between zero and one. The probability of any particular outcome is zero, but it’s zero reached in a very particular way (basically, 1/Q, where Q is the infinite cardinality of the rational numbers), because all the infinitely many zeroes have to sum to one.
Getting more precise than that would require getting into the formal underpinnings of calculus and limits.
I will say it is not strictly, formally true that rolling a 7 on a normally-numbered six sided die is zero. That’s a rounding convention we use. There is a non-zero, not-technically-infinitesimal probably of the atoms in the die randomly ended up in a configuration, during the roll, where one side has seven dots. Are there classes of problem where the difference between an infinitesimal and an extremely small but finite number can matter? Sure. But the difference usually doesn’t matter in practice.
So if I’m interpreting it correctly, in the general case Bayesians can reasonably assign a probability of 0 to an event that can actually happen, and a probability 1 event under Bayes is not equal to the event always happening.
Yes, but see @Amarko ’s reply and corrections, below. The examples where this ends up being possible are all theoretical and generally involve some form of infinite or infinitesimal quantities.
I gave 2 examples of probability 0 or 1 plausibly occuring in real life, and 1 of them relies on the uniform distribution of all real numbers, where if you pick one of them randomly, no matter which number you pick, it always has probability 0, and the set of Turing Machines that have a decidable halting problem, which has probability 1, but you can’t extend them into never getting a real number constant/always deciding the halting problem.
It is not at all clear that it is possible in reality to randomly select a real number without a process that can make an infinite number of choices in finite time. Similarly, any reasoning about Turing machines has to acknowledge that no real, physical system actually instantiates one in the sense of having an infinite tape and never making an error. We can approach/approximate these examples, but that just means we end up with probabilities that are small-but-finite, not 0 or 1
This is not quite accurate. You can’t uniformly pick a random rational number from 0 to 1, because there are countably many such numbers, and any probability distribution you assign will have to add up to 1. Every probability distribution on this set assigns a nonzero probability to every number.
You can have a uniform distribution on an uncountable set, such as the real numbers between 0 and 1, but since you can’t pick an arbitrary element of an uncountable set in the real world this is theoretical rather than a real-world issue.
As far as I know, any mathematical case in which something with probability 0 can happen does not actually occur in the real world in a way that we can observe.
As far as I know, any mathematical case in which something with probability 0 can happen does not actually occur in the real world in a way that we can observe.
Here’s one example:
What’s the probability that our physical constants are what they are, especially the constants that seem tuned to life?
The answer is if the constants are arbitrary real numbers, the answer is probability 0, and this applies no matter what number you pick.
This is how we can defuse the fine-tuning argument, that the cosmos’s constants have improbable values that seem tuned for life, since any other constant has probability 0, no matter whether it was able to sustain life or not:
The question to ask is, what is the measure of the space of physical constants compatible with life? Although that requires some prior probability measure on the space of all hypothetical values of the constants. But the constants are mostly real numbers, and there is no uniform distribution on the reals.
Thanks, I didn’t realize that! It does make sense now that I think about it. I think if you replace the rationals with the reals in my theoretical example, the rest still works?
And yes, I agree about in the real world. Probabilities 0 and 1 are limits you can approach, but only reach in theory.
Strictly speaking, no, it does not mean that.
Pick a random rational number between zero and one. The probability of any particular outcome is zero, but it’s zero reached in a very particular way (basically, 1/Q, where Q is the infinite cardinality of the rational numbers), because all the infinitely many zeroes have to sum to one.
Getting more precise than that would require getting into the formal underpinnings of calculus and limits.
I will say it is not strictly, formally true that rolling a 7 on a normally-numbered six sided die is zero. That’s a rounding convention we use. There is a non-zero, not-technically-infinitesimal probably of the atoms in the die randomly ended up in a configuration, during the roll, where one side has seven dots. Are there classes of problem where the difference between an infinitesimal and an extremely small but finite number can matter? Sure. But the difference usually doesn’t matter in practice.
Thanks for the answer.
So if I’m interpreting it correctly, in the general case Bayesians can reasonably assign a probability of 0 to an event that can actually happen, and a probability 1 event under Bayes is not equal to the event always happening.
Is this correctly interpreted?
Yes, but see @Amarko ’s reply and corrections, below. The examples where this ends up being possible are all theoretical and generally involve some form of infinite or infinitesimal quantities.
I gave 2 examples of probability 0 or 1 plausibly occuring in real life, and 1 of them relies on the uniform distribution of all real numbers, where if you pick one of them randomly, no matter which number you pick, it always has probability 0, and the set of Turing Machines that have a decidable halting problem, which has probability 1, but you can’t extend them into never getting a real number constant/always deciding the halting problem.
It is not at all clear that it is possible in reality to randomly select a real number without a process that can make an infinite number of choices in finite time. Similarly, any reasoning about Turing machines has to acknowledge that no real, physical system actually instantiates one in the sense of having an infinite tape and never making an error. We can approach/approximate these examples, but that just means we end up with probabilities that are small-but-finite, not 0 or 1
This is not quite accurate. You can’t uniformly pick a random rational number from 0 to 1, because there are countably many such numbers, and any probability distribution you assign will have to add up to 1. Every probability distribution on this set assigns a nonzero probability to every number.
You can have a uniform distribution on an uncountable set, such as the real numbers between 0 and 1, but since you can’t pick an arbitrary element of an uncountable set in the real world this is theoretical rather than a real-world issue.
As far as I know, any mathematical case in which something with probability 0 can happen does not actually occur in the real world in a way that we can observe.
Here’s one example:
What’s the probability that our physical constants are what they are, especially the constants that seem tuned to life?
The answer is if the constants are arbitrary real numbers, the answer is probability 0, and this applies no matter what number you pick.
This is how we can defuse the fine-tuning argument, that the cosmos’s constants have improbable values that seem tuned for life, since any other constant has probability 0, no matter whether it was able to sustain life or not:
https://en.wikipedia.org/wiki/Fine-tuned_universe
The question to ask is, what is the measure of the space of physical constants compatible with life? Although that requires some prior probability measure on the space of all hypothetical values of the constants. But the constants are mostly real numbers, and there is no uniform distribution on the reals.
Thanks, I didn’t realize that! It does make sense now that I think about it. I think if you replace the rationals with the reals in my theoretical example, the rest still works?
And yes, I agree about in the real world. Probabilities 0 and 1 are limits you can approach, but only reach in theory.