A true Bayesian has unlimited information handling ability.
I think I see that—because if it didn’t, then not all of its probabilities would be properly updated, so its degrees of belief wouldn’t have the relations implied by probability theory, so it wouldn’t be a true Bayesian. Right?
Yes, one generally ignores the cost of making these computations. One might try to take it into account, but then one is ignoring the cost of doing that computation, etc. Historically, the “Bayesian revolution” needed computers before it could happen.
And, I notice, it has only gone as far as the computers allow. “True Bayesians” also have universal priors, that assign non-zero probability density to every logically possible hypothesis. Real Bayesian statisticians never do this; all those I have read deny that it is possible.
And, I notice, it has only gone as far as the computers allow. “True Bayesians” also have universal priors, that assign non-zero probability density to every logically possible hypothesis. Real Bayesian statisticians never do this; all those I have read deny that it is possible.
It is impossible, even in principal. The only way to have universal priors over all computable universes is if you have access to a source of hypercomputation, but that would mean the universe isn’t computable so the truth still isn’t in your prior set.
Yes, in this technical sense.
A true Bayesian has unlimited information handling ability.
I think I see that—because if it didn’t, then not all of its probabilities would be properly updated, so its degrees of belief wouldn’t have the relations implied by probability theory, so it wouldn’t be a true Bayesian. Right?
Yes, one generally ignores the cost of making these computations. One might try to take it into account, but then one is ignoring the cost of doing that computation, etc. Historically, the “Bayesian revolution” needed computers before it could happen.
And, I notice, it has only gone as far as the computers allow. “True Bayesians” also have universal priors, that assign non-zero probability density to every logically possible hypothesis. Real Bayesian statisticians never do this; all those I have read deny that it is possible.
It is impossible, even in principal. The only way to have universal priors over all computable universes is if you have access to a source of hypercomputation, but that would mean the universe isn’t computable so the truth still isn’t in your prior set.
Is that written up as a theorem anywhere?
That depends on how one wants to formalize it.