What I’m saying is that effective techniques for reasoning non-probablistically derive their effectiveness from their similarity to Bayesian reasoning. In some special cases, they are identical to Bayesian reasoning, but in other cases, Bayes is superior. This is because Bayes is proved, by theorem, to be Right. Not Unbiased or Robust, but Right.
So I’m not saying it’s a subset, I’m saying its similar to. One way to be similar to something is to copy some but not all of it, which is taking a subset plus adding other stuff. This is almost the same concept, obviously.
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Using the two logics I know best:
Classical propositional logic is Bayesian logic for probabilities of 1 and 0.
All results of Bayesian logic can be derived by propositional logic and certain additional assumptions.
But these are qualitatively different. I don’t know how to describe this difference—which one is fundamental? I think both are, in different ways.
What’s missing is how you use it in real-world problems, I think. Because it is often the case that one formal system is contained in another, and the other is contained in the one. But formal systems are only useful if they model reality, or serve some other function. Classical logic is used more than intuitionistic logic—is this because it’s more useful? (Classical logic is more useful than Bayesian in the world of mathematics, but less in the real world, I think.)
What I’m saying is that effective techniques for reasoning non-probablistically derive their effectiveness from their similarity to Bayesian reasoning. In some special cases, they are identical to Bayesian reasoning, but in other cases, Bayes is superior. This is because Bayes is proved, by theorem, to be Right. Not Unbiased or Robust, but Right.
http://lesswrong.com/lw/mt/beautiful_probability/ seems like the appropriate lesswrong post.
So I’m not saying it’s a subset, I’m saying its similar to. One way to be similar to something is to copy some but not all of it, which is taking a subset plus adding other stuff. This is almost the same concept, obviously. ` Using the two logics I know best: Classical propositional logic is Bayesian logic for probabilities of 1 and 0. All results of Bayesian logic can be derived by propositional logic and certain additional assumptions.
But these are qualitatively different. I don’t know how to describe this difference—which one is fundamental? I think both are, in different ways.
What’s missing is how you use it in real-world problems, I think. Because it is often the case that one formal system is contained in another, and the other is contained in the one. But formal systems are only useful if they model reality, or serve some other function. Classical logic is used more than intuitionistic logic—is this because it’s more useful? (Classical logic is more useful than Bayesian in the world of mathematics, but less in the real world, I think.)
I need to think more about this...