Remember that in Bayesian epistemology, probabilities represent our state of knowledge, so as you pointed out, the simplest hypothesis that fits the data so far may not be the true one because we haven’t seen all of the data. But it is necessarily our best guess because of the conjunction rule.
You are going to have to expand on this. I don’t see how the conjunction rule implies that simpler hypotheses are in general more probable. This is true if we have two hypotheses where one is X and the other is “X and Y” but that’s not how people generally apply this sort of thing. For example, I might have a sequence of numbers that for the first 10,000 terms has the nth term as the nth prime number. One hypothesis is that the nth term is always the nth prime number. But I could have as another hypothesis some high degree polynomial that matches the first 10,000 primes. That’s clearly more complicated. But one can’t use conjunction to argue that it is less likely.
Imagine that I have some set of propositions, A through Z, and I don’t know the probabilities of any of these. Now let’s say I’m using these propositions to explain some experimental result—since I would have uniform priors for A through Z, it follows that an explanation like “M did it” is more probable than “A and B did it,” which in turn is more probable than “G and P and H did it.”
Yes, I agree with you there. But this is much weaker than any general form of Occam. See my example with primes. What we want to say in some form of Occam approach is much stronger than what you can get from simply using the conjunction argument.
You are going to have to expand on this. I don’t see how the conjunction rule implies that simpler hypotheses are in general more probable. This is true if we have two hypotheses where one is X and the other is “X and Y” but that’s not how people generally apply this sort of thing. For example, I might have a sequence of numbers that for the first 10,000 terms has the nth term as the nth prime number. One hypothesis is that the nth term is always the nth prime number. But I could have as another hypothesis some high degree polynomial that matches the first 10,000 primes. That’s clearly more complicated. But one can’t use conjunction to argue that it is less likely.
Imagine that I have some set of propositions, A through Z, and I don’t know the probabilities of any of these. Now let’s say I’m using these propositions to explain some experimental result—since I would have uniform priors for A through Z, it follows that an explanation like “M did it” is more probable than “A and B did it,” which in turn is more probable than “G and P and H did it.”
Yes, I agree with you there. But this is much weaker than any general form of Occam. See my example with primes. What we want to say in some form of Occam approach is much stronger than what you can get from simply using the conjunction argument.