You mean proven given some assumptions about what an epistemology should be, right?
Yes, I stated my assumptions in the sentence, though I may have missed some.
We always face this kind of problem to do much of anything. How do we figure out when our theories apply?
This comes back to the distinction between one complete theory that fully specifies the universe and a set of theories that are considered to be one because we are only looking at a certain domain. In the former case, the domain of applicability is everywhere. In the latter, we have a probability distribution that tells us how likely it is to fail in every domain. So, this kind of thing is all there in the math.
I replied to that. Repeating: if you increase the probability of infinitely many theories, the problem of figuring out a good theory is not solved. So that is not all you need.
What do you mean by ‘a good theory’. Bayesian never select one theory as ‘good’ as follow that; we always consider the possibility of being wrong. When theories have higher probability than others, I guess you could call them good. I don’t see why this is hard; just calculate P(H | E) for all the theories and give more weight to the more likely ones when making decisions.
Further, I’m still waiting on an adequate answer about what support is (inductive or otherwise) and how it differs from consistency.
Evidence supports a hypothesis if P(H | E) > P(H). Two statements, A, B, are consistent if ¬(A&B → ⊥). I think I’m missing something.
Evidence supports a hypothesis if P(H | E) > P(H). Two statements, A, B, are consistent if ¬(A&B → ⊥). I think I’m missing something.
Let’s consider only theories which make all their predictions with 100% probability for now. And theories which cover everything.
Then:
If H and E are consistent, then it follows that P(H | E) > P(H).
For any given E, consider how much greater the probability of H is, for all consistent H. That amount is identical for all H considered.
We can put all the Hs in two categories: the consistent ones which gain equal probability, and the inconsistent ones for which P(H|E) = 0. (Assumption warning: we’re relying on getting it right which H are consistent with which E.)
This means:
1) consistency and support coincide.
2) there are infinitely many equally supported theories. There are only and exactly two amounts of support that any theory has given all current evidence, one of which is 0.
3) The support concept plays no role in helping us distinguish between the theories with more than 0 support.
4) The support concept can be dropped entirely because it has no use at all. The consistency concept does everything
5) All mention of probability can be dropped too, since it wasn’t doing anything.
6) And we still have the main problem of epistemology left over, which is dealing with the theories that aren’t refuted by evidence
Similar arguments can be made without my initial assumptions/restrictions. For example introducing theories that make predictions with less than 100% probability will not help you because they are going to have lower probability than theories which make the same predictions with 100% probability.
For any given E, consider how much greater the probability of H is, for all consistent H. That amount is identical for all H considered.
Well the ratio is the same, but that’s probably what you meant.
5) All mention of probability can be dropped too, since it wasn’t doing anything.
6) And we still have the main problem of epistemology left over, which is dealing with the theories that aren’t refuted by evidence
Have a prior. This reintroduces probabilities and deals with the remaining theories. You will converge on the right theory eventually no matter what your prior is. Of course, that does not mean that all priors are equally rational.
If they all have the same prior probability, then their probabilities are the same and stay that way. If you use a prior which arbitrarily (in my view) gives some things higher prior probabilities in a 100% non-evidence-based way, I object to that, and it’s a separate issue from support.
How does having a prior save the concept of support? Can you give an example? Maybe the one here, currently near the bottom:
If they all have the same prior probability, then their probabilities are the same and stay that way.
Well shouldn’t they? If you look at it from the perspective of making decisions rather than finding one right theory, it’s obvious that they are equiprobable and this should be recognized.
If you use a prior which arbitrarily (in my view) gives some things higher prior probabilities in a 100% non-evidence-based way, I object to that, and it’s a separate issue from support.
Solomonoff does not give “some things higher prior probabilities in a 100% non-evidence-based way”. All hypotheses have the same probability, many just make similar predictions.
Yes, I stated my assumptions in the sentence, though I may have missed some.
This comes back to the distinction between one complete theory that fully specifies the universe and a set of theories that are considered to be one because we are only looking at a certain domain. In the former case, the domain of applicability is everywhere. In the latter, we have a probability distribution that tells us how likely it is to fail in every domain. So, this kind of thing is all there in the math.
What do you mean by ‘a good theory’. Bayesian never select one theory as ‘good’ as follow that; we always consider the possibility of being wrong. When theories have higher probability than others, I guess you could call them good. I don’t see why this is hard; just calculate P(H | E) for all the theories and give more weight to the more likely ones when making decisions.
Evidence supports a hypothesis if P(H | E) > P(H). Two statements, A, B, are consistent if ¬(A&B → ⊥). I think I’m missing something.
Let’s consider only theories which make all their predictions with 100% probability for now. And theories which cover everything.
Then:
If H and E are consistent, then it follows that P(H | E) > P(H).
For any given E, consider how much greater the probability of H is, for all consistent H. That amount is identical for all H considered.
We can put all the Hs in two categories: the consistent ones which gain equal probability, and the inconsistent ones for which P(H|E) = 0. (Assumption warning: we’re relying on getting it right which H are consistent with which E.)
This means:
1) consistency and support coincide.
2) there are infinitely many equally supported theories. There are only and exactly two amounts of support that any theory has given all current evidence, one of which is 0.
3) The support concept plays no role in helping us distinguish between the theories with more than 0 support.
4) The support concept can be dropped entirely because it has no use at all. The consistency concept does everything
5) All mention of probability can be dropped too, since it wasn’t doing anything.
6) And we still have the main problem of epistemology left over, which is dealing with the theories that aren’t refuted by evidence
Similar arguments can be made without my initial assumptions/restrictions. For example introducing theories that make predictions with less than 100% probability will not help you because they are going to have lower probability than theories which make the same predictions with 100% probability.
Well the ratio is the same, but that’s probably what you meant.
Have a prior. This reintroduces probabilities and deals with the remaining theories. You will converge on the right theory eventually no matter what your prior is. Of course, that does not mean that all priors are equally rational.
If they all have the same prior probability, then their probabilities are the same and stay that way. If you use a prior which arbitrarily (in my view) gives some things higher prior probabilities in a 100% non-evidence-based way, I object to that, and it’s a separate issue from support.
How does having a prior save the concept of support? Can you give an example? Maybe the one here, currently near the bottom:
http://lesswrong.com/lw/54u/bayesian_epistemology_vs_popper/3urr?context=3
Well shouldn’t they? If you look at it from the perspective of making decisions rather than finding one right theory, it’s obvious that they are equiprobable and this should be recognized.
Solomonoff does not give “some things higher prior probabilities in a 100% non-evidence-based way”. All hypotheses have the same probability, many just make similar predictions.