I’ve seen that article before, but can’t quite understand it. Is there really a use for mixed sentences like “the probability that the probability that all ravens are black is 0.5 is 0.5”? It seems like both quantifiers and meta-probabilities are unnecessary, I can say all I want just by having a prior over states of the world with all its ravens. Relationships among multiple objects get folded into that as well.
Sure, but you can’t actually hold the probability vector over all states with ravens. So you move up a level and summarize that set of probabilities to a smaller (and less precise) set.
All uncertainty is map, not territory. Anytime you are using probability, you’re acknowledging that you’re a limited calculator that cannot hold the complete state of the universe. If you could, you wouldn’t need probability, you’d actually know the thing.
Meta-models are useful when specific models get cumbersome. Likewise meta-probability.
You don’t need meta-probability to compress priors. For example, a uniform prior on [0,1] talks about an uncountable set of events, but its description is tiny and doesn’t use meta-probabilities.
It doesn’t have to be infinitely complex. Let’s say there are only ten ravens and ten crows, each of which can be black or white. Chapman says I can’t talk about them using probability theory because there are two kinds of objects, so I need meta-probabilities and quantifiers and whatnot. But I don’t need any of that stuff, it’s enough to have a prior over possible worlds, which would be finite and rather small.
That amounts to saying that Bayes works in finite, restricted cases, which no one is disputing. The thing is that you scheme doesn’t work in the general case.
I can say all I want just by having a prior over states of the world with all its ravens
No, you can’t. Not in practice.
It’s the same deal as with AIXI—quite omnipotent in theory, can’t do much of anything in reality. Take a real-life problem and show me your prior over all the states of the world.
All priors are over the state of the world, just coarse-grained :-) So any practical application of Bayesian statistics should suffice for your request.
So any practical application of Bayesian statistics should suffice for your request.
Any practical application does not give me an opportunity to “say all I want just by having a prior over states of the world” because it doesn’t involve such a prior. A practical application sets out a model with some parameters and invites me to specify (preferably in a neat analytical form) the prior for these parameters.
I’ve seen that article before, but can’t quite understand it. Is there really a use for mixed sentences like “the probability that the probability that all ravens are black is 0.5 is 0.5”? It seems like both quantifiers and meta-probabilities are unnecessary, I can say all I want just by having a prior over states of the world with all its ravens. Relationships among multiple objects get folded into that as well.
Sure, but you can’t actually hold the probability vector over all states with ravens. So you move up a level and summarize that set of probabilities to a smaller (and less precise) set.
All uncertainty is map, not territory. Anytime you are using probability, you’re acknowledging that you’re a limited calculator that cannot hold the complete state of the universe. If you could, you wouldn’t need probability, you’d actually know the thing.
Meta-models are useful when specific models get cumbersome. Likewise meta-probability.
You don’t need meta-probability to compress priors. For example, a uniform prior on [0,1] talks about an uncountable set of events, but its description is tiny and doesn’t use meta-probabilities.
And it’s a special case.
How can you “have” an infinitely complex prior?
It doesn’t have to be infinitely complex. Let’s say there are only ten ravens and ten crows, each of which can be black or white. Chapman says I can’t talk about them using probability theory because there are two kinds of objects, so I need meta-probabilities and quantifiers and whatnot. But I don’t need any of that stuff, it’s enough to have a prior over possible worlds, which would be finite and rather small.
Only you need to keep switching priors to deal with one finite and small problem after another. Whatever that is, it is not strong Bayes.
That amounts to saying that Bayes works in finite, restricted cases, which no one is disputing. The thing is that you scheme doesn’t work in the general case.
No, you can’t. Not in practice.
It’s the same deal as with AIXI—quite omnipotent in theory, can’t do much of anything in reality. Take a real-life problem and show me your prior over all the states of the world.
All priors are over the state of the world, just coarse-grained :-) So any practical application of Bayesian statistics should suffice for your request.
Any practical application does not give me an opportunity to “say all I want just by having a prior over states of the world” because it doesn’t involve such a prior. A practical application sets out a model with some parameters and invites me to specify (preferably in a neat analytical form) the prior for these parameters.