I still think that these “devastating” problems have been solved in the first chapters of Jaynes’ book.
“So if the green premises are evidence that the next emerald will be green, why aren’t the grue premises evidence for the next emerald being grue?”
Because the first green emeralds are no evidence that the next will be green.
Let’s translate the problem differently: I write a program that shows colored dot on the screen. The first n dot are green. What is the probability that the next dot will be green? If those are your only information, you simply cannot tell. By the principle of indifference, the probability is 1/(the number of possible colors represented on the screen). I could have coded the program with a counter so that it shows only pink dots after the billionth green dot, and you would never know.
If seeing n green emerald makes you raise the probability that the next emerald is green, you are making very specific and yet hidden assumptions.
Let’s add for example a different information: you are extracting emeralds from an urn that contains 99 green emerald and only one blue emerald. Then the probability that the next emerald is green, after having seen n<99 emerald is lower, not higher.
That said, let’s talk about real emerald. In this case, we know that we are extracting emerald from an urn (the Earth) that by the time of the execution of the experiment, contains only N emerald. We know something about the process of production of the emeralds, and we also know that the process that produces blue emerald is extremely unlikely.
What does the green hypotheis say? It assumes that the blue emerald process didn’t happen. So it is correct to say that the next emerald will be green (with probability one, however. It doesn’t increase).
What does the grue hypothesis say? That the blue process did infact happened, so whenever we observe a green emerald the probability of the next being blue (and so the probability of it being grue) increases (and similarly, the probability of it being green decreases accordingly).
The paradox then doesn’t happen because green and grue have very distinct implicit assumptions.
If we can’t justify inferring probability distributions over random variables based on their previous results, we cannot justify a single bit of natural science.
I still think that these “devastating” problems have been solved in the first chapters of Jaynes’ book.
Because the first green emeralds are no evidence that the next will be green.
Let’s translate the problem differently: I write a program that shows colored dot on the screen. The first n dot are green. What is the probability that the next dot will be green? If those are your only information, you simply cannot tell. By the principle of indifference, the probability is 1/(the number of possible colors represented on the screen). I could have coded the program with a counter so that it shows only pink dots after the billionth green dot, and you would never know.
If seeing n green emerald makes you raise the probability that the next emerald is green, you are making very specific and yet hidden assumptions.
Let’s add for example a different information: you are extracting emeralds from an urn that contains 99 green emerald and only one blue emerald. Then the probability that the next emerald is green, after having seen n<99 emerald is lower, not higher.
That said, let’s talk about real emerald. In this case, we know that we are extracting emerald from an urn (the Earth) that by the time of the execution of the experiment, contains only N emerald. We know something about the process of production of the emeralds, and we also know that the process that produces blue emerald is extremely unlikely.
What does the green hypotheis say? It assumes that the blue emerald process didn’t happen. So it is correct to say that the next emerald will be green (with probability one, however. It doesn’t increase).
What does the grue hypothesis say? That the blue process did infact happened, so whenever we observe a green emerald the probability of the next being blue (and so the probability of it being grue) increases (and similarly, the probability of it being green decreases accordingly).
The paradox then doesn’t happen because green and grue have very distinct implicit assumptions.
This is seriously wrong.