Abnormalised sampling? Probability theory talks about sampling for probability distributions, i.e. normalized measures. However, non-normalized measures abound: weighted automata, infra-stuff, uniform priors on noncompact spaces, wealth in logical-inductor esque math, quantum stuff?? etc.
Most of probability theory constructions go through just for arbitrary measures, doesn’t need the normalization assumption. Except, crucially, sampling.
What does it even mean to sample from a non-normalized measure? What is unnormalized abnormal sampling?
I don’t know.
Infra-sampling has an interpretation of sampling from a distribution made by a demonic choice. I don’t have good interpretations for other unnormalized measures.
Concrete question: is there a law of large numbers for unnormalized measures?
Let f be a measureable function and m a measure. Then the expectation value is defined Em(f)=∫fdm. A law of large numbers for unnormalized measure would have to say something about repeated abnormal sampling.
Abnormalised sampling?
Probability theory talks about sampling for probability distributions, i.e. normalized measures. However, non-normalized measures abound: weighted automata, infra-stuff, uniform priors on noncompact spaces, wealth in logical-inductor esque math, quantum stuff?? etc.
Most of probability theory constructions go through just for arbitrary measures, doesn’t need the normalization assumption. Except, crucially, sampling.
What does it even mean to sample from a non-normalized measure? What is
unnormalizedabnormal sampling?I don’t know.
Infra-sampling has an interpretation of sampling from a distribution made by a demonic choice. I don’t have good interpretations for other unnormalized measures.
Concrete question: is there a law of large numbers for unnormalized measures?
Let f be a measureable function and m a measure. Then the expectation value is defined Em(f)=∫fdm. A law of large numbers for unnormalized measure would have to say something about repeated abnormal sampling.
I have no real ideas. Curious to learn more.