Try rephrasing your first paragraph when the quantity of interest is not a frequency but, say, Avogadro’s number, and you’re Jean Perrin trying to determine exactly what that number is.
A frequentist would take a probability model for the data you’re generating and give you a confidence interval. A billion scientists repeat your experiments, getting their own data and their own intervals. Among those intervals, the proportion that contain the true value of Avogadro’s number is equal to the confidence (up to sampling error).
A Bayesian would take the same probability model, plus a prior, and combine them using Bayes. Each scientist may have her own prior, and posterior calibration is only guaranteed if (i) all the priors taken as a group were calibrated, or, (ii) everyone is using the matching prior if it exists (these are typically improper, so prior calibration cannot be calculated).
Try rephrasing your first paragraph when the quantity of interest is not a frequency but, say, Avogadro’s number, and you’re Jean Perrin trying to determine exactly what that number is.
A frequentist would take a probability model for the data you’re generating and give you a confidence interval. A billion scientists repeat your experiments, getting their own data and their own intervals. Among those intervals, the proportion that contain the true value of Avogadro’s number is equal to the confidence (up to sampling error).
A Bayesian would take the same probability model, plus a prior, and combine them using Bayes. Each scientist may have her own prior, and posterior calibration is only guaranteed if (i) all the priors taken as a group were calibrated, or, (ii) everyone is using the matching prior if it exists (these are typically improper, so prior calibration cannot be calculated).