If Bayesian epistemology is mathematically proven, why have I been told in my discussions here various things such as: there is a regress problem which isn’t fully solved (Yudkowsky says so), that circular arguments for induction are correction, that foundationalism is correct, been linked to articles to make Bayesian points and told they have good arguments with only a little hand waving, and so on? And I’ve been told further research is being done.
It is proven that the correct epistemology, meaning one that is necessary to achieve general goals, is isomorphic to Bayesianism with some prior (for beings that know all math). What that prior is requires more work. While the constraint of knowing all math is extremely unrealistic, do you agree that the theory of what knowledge would be had in such situations is a useful guide to action until we have a more general theory. Popperian epistemology cannot tell me how much money to bet at what odds for or against P = NP any more than Bayesian epistemology can, but at least Bayesian epistemology set this as a goal.
it is instrumentalist (has not understood the value of explanatory knowledge)
oracles have user interfaces and entering questions isn’t just free and trivial, and did not address the issue that physical reality is a predictive oracle meeting all the specified characteristics of the alien oracle
This is all based on our limited mathematical ability. A theory does have an advantage over an oracle or the reality-oracle: we can read it. Would you agree that all the benefits of a theory come from this plus knowing all math. The difference is one of mathematical knowledge, not of physical knowledge. How does Popper help with this? Are there guidelines for what ‘equivalent’ formulations of a theory are mathematically better? If so, this is something that Bayesianism does not try to cover, so this may have value. However, this is unrelated to the question of the validity of “don’t assign probabilities to theories”.
inductivist (induction is refuted)
I thought I addressed this but, to recap:
p(h, eb) > p(h, b) [bottom left of page 1]
That (well and how much bigger) is all I need to make decisions.
All this means: that factor that contains all of h that does not follow deductively from e is strongly countersupported by e.
So what? I already have my new probabilities.
[T]he calculus of probability reveals that probabilistic support cannot be inductive support.
What is induction if not the calculation of new probabilities for hypotheses? Should I care about these ‘inductive truths’ that Popper disproves the existence of? I already have an algorithm to calculate the best action to take. It seems like Bayesianism isn’t inductivist by Popper’s definition.
moral knowledge
I’d like to be sure that we are using the same definitions of our terms, so please give an example.
You mean proven given some assumptions about what an epistemology should be, right?
Would you agree that all the benefits of a theory come from this [can read it] plus knowing all math.
No. We need explanations to understand the world. In real life, is only when we have explanations that we can make good predictions at all. For example, suppose you have a predictive theory about dice and you want to make bets. I chose that example intentionally to engage with areas of your strength. OK, now you face the issue: does a particular real world situation have the correct attributes for my predictive theory to apply? You have to address that to know if your predictions will be correct or not. We always face this kind of problem to do much of anything. How do we figure out when our theories apply? We come up with explanations about what kinds of situations they apply to, and what situation we are in, and we then come up with explanations about why we think we are/aren’t in the right kind of situation, and we use critical argument to improve these explanations. Bayesian Epistemology does not address all this.
p(h, eb) > p(h, b) [bottom left of page 1]
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.
Further, I’m still waiting on an adequate answer about what support is (inductive or otherwise) and how it differs from consistency.
I gave examples of moral knowledge in another comment to you. Morality is knowledge about how to live, what is a good life. e.g. murder is immoral.
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.
It is proven that the correct epistemology, meaning one that is necessary to achieve general goals, is isomorphic to Bayesianism with some prior (for beings that know all math). What that prior is requires more work. While the constraint of knowing all math is extremely unrealistic, do you agree that the theory of what knowledge would be had in such situations is a useful guide to action until we have a more general theory. Popperian epistemology cannot tell me how much money to bet at what odds for or against P = NP any more than Bayesian epistemology can, but at least Bayesian epistemology set this as a goal.
This is all based on our limited mathematical ability. A theory does have an advantage over an oracle or the reality-oracle: we can read it. Would you agree that all the benefits of a theory come from this plus knowing all math. The difference is one of mathematical knowledge, not of physical knowledge. How does Popper help with this? Are there guidelines for what ‘equivalent’ formulations of a theory are mathematically better? If so, this is something that Bayesianism does not try to cover, so this may have value. However, this is unrelated to the question of the validity of “don’t assign probabilities to theories”.
I thought I addressed this but, to recap:
That (well and how much bigger) is all I need to make decisions.
So what? I already have my new probabilities.
What is induction if not the calculation of new probabilities for hypotheses? Should I care about these ‘inductive truths’ that Popper disproves the existence of? I already have an algorithm to calculate the best action to take. It seems like Bayesianism isn’t inductivist by Popper’s definition.
I’d like to be sure that we are using the same definitions of our terms, so please give an example.
You mean proven given some assumptions about what an epistemology should be, right?
No. We need explanations to understand the world. In real life, is only when we have explanations that we can make good predictions at all. For example, suppose you have a predictive theory about dice and you want to make bets. I chose that example intentionally to engage with areas of your strength. OK, now you face the issue: does a particular real world situation have the correct attributes for my predictive theory to apply? You have to address that to know if your predictions will be correct or not. We always face this kind of problem to do much of anything. How do we figure out when our theories apply? We come up with explanations about what kinds of situations they apply to, and what situation we are in, and we then come up with explanations about why we think we are/aren’t in the right kind of situation, and we use critical argument to improve these explanations. Bayesian Epistemology does not address all this.
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.
Further, I’m still waiting on an adequate answer about what support is (inductive or otherwise) and how it differs from consistency.
I gave examples of moral knowledge in another comment to you. Morality is knowledge about how to live, what is a good life. e.g. murder is immoral.
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.