I would argue that the properties Bayesians think are important are important, and the properties frequentists think are important are also important.
This favors a view in which we track both subjective and objective probability.
Bayesians usually have to track parameters (EG in their conditional probability tables) anyway. These are essentially “objective chances”!
Frequentists have to make decisions, at which point they often have to go beyond what’s established as a matter of science and make some judgement calls. Bayesian decision theory is a theory of how to do this.
As I’ve argued elsewhere, Bayesian updates have a convergence problem. This can be fixed with frequentist-leaning ideas. Similarly, Bayesian probabilities are not guaranteed to be calibrated, but this can also be addressed with some frequentist-leaning ideas. And similarly again, Bayesian priors have a realizability problem, which can be fixed with frequentist-leaning ideas.
Frequentist probability is objective because it defined in terms of falsifiable real-world observations. An objective definition of probability can be used to resolve disagreements between scientists. A subjective definition cannot. That’s why Frequentist probability is used as the foundation of science and Bayesian probability isn’t.
Frequentists say that probabilities are external to humans, and hence, cannot be definitively measured. In my experience, frequentists are under no illusion that objective probabilities can be known. Frequentist procedures can get unlucky and reject a true hypothesis.
The definition—probabilities are limiting frequencies of repeated experiments—does not in any way imply that results can’t cluster in weird ways in the sequence of experiments. For example, the statement that the frequency of an event limits to 0 is totally consistent with the first billion experiments displaying frequency 1. You need the (unjustified and unjustifiable) IID assumption to rule this out even probabilistically.
(I’ll grant that frequentists have some great ideas, but imho the IID assumption is not one of them.)
Why not both?
I would argue that the properties Bayesians think are important are important, and the properties frequentists think are important are also important.
This favors a view in which we track both subjective and objective probability.
Bayesians usually have to track parameters (EG in their conditional probability tables) anyway. These are essentially “objective chances”!
Frequentists have to make decisions, at which point they often have to go beyond what’s established as a matter of science and make some judgement calls. Bayesian decision theory is a theory of how to do this.
As I’ve argued elsewhere, Bayesian updates have a convergence problem. This can be fixed with frequentist-leaning ideas. Similarly, Bayesian probabilities are not guaranteed to be calibrated, but this can also be addressed with some frequentist-leaning ideas. And similarly again, Bayesian priors have a realizability problem, which can be fixed with frequentist-leaning ideas.
Frequentists say that probabilities are external to humans, and hence, cannot be definitively measured. In my experience, frequentists are under no illusion that objective probabilities can be known. Frequentist procedures can get unlucky and reject a true hypothesis.
The definition—probabilities are limiting frequencies of repeated experiments—does not in any way imply that results can’t cluster in weird ways in the sequence of experiments. For example, the statement that the frequency of an event limits to 0 is totally consistent with the first billion experiments displaying frequency 1. You need the (unjustified and unjustifiable) IID assumption to rule this out even probabilistically.
(I’ll grant that frequentists have some great ideas, but imho the IID assumption is not one of them.)