“But frequentists emphatically are not talking about individual tosses. They are talking about infinitely repeated tosses.”
These infinite sequences never exist, and very often they don’t even exist approximately. We only observe finite numbers of events. I think this is one of the things Jaynes had in mind when he talked about the proper handling of infinities—you should start by analyzing the finite case, and look for a well-defined limit as n increases without bound. Unfortunately, frequentist statistics starts with the limit at infinity.
As an example of how these limiting frequencies taken over infinite sequences often make no sense in real-world situations, consider statistical models of human language, such as are used in automatic speech recognition. Such models assign a prior probability to each possible utterance a person could make. What does it mean, from a frequentist standpoint, to say that there is a probability of 1e-100 that a person will say “The tomatoe flew dollars down the pipe”? There haven’t been 1e100 separate utterances by all human beings in all of human history, so how could a probability of 1e-100 possibly correspond to some sort of long-run frequency?
“But frequentists emphatically are not talking about individual tosses. They are talking about infinitely repeated tosses.”
These infinite sequences never exist, and very often they don’t even exist approximately. We only observe finite numbers of events. I think this is one of the things Jaynes had in mind when he talked about the proper handling of infinities—you should start by analyzing the finite case, and look for a well-defined limit as n increases without bound. Unfortunately, frequentist statistics starts with the limit at infinity.
As an example of how these limiting frequencies taken over infinite sequences often make no sense in real-world situations, consider statistical models of human language, such as are used in automatic speech recognition. Such models assign a prior probability to each possible utterance a person could make. What does it mean, from a frequentist standpoint, to say that there is a probability of 1e-100 that a person will say “The tomatoe flew dollars down the pipe”? There haven’t been 1e100 separate utterances by all human beings in all of human history, so how could a probability of 1e-100 possibly correspond to some sort of long-run frequency?