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
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