From Relative Frequencies to Bayesian Probabilities

Abstract

It is uncontroversial that evidence regarding frequencies should constrain probabilities or degrees of belief. What is controversial is the question of how this should be done. The random-worlds method provides some insight on this question. However, the method by itself faces problems in accounting for rational inferences from samples and in accommodating the uncertainty that agents occasionally have about relevant relative frequencies. One potential response to these problems is to seek alternative probability measures to accommodate such inferences and uncertainty. After surveying two such measures and various problems for them, I find this response wanting. I then offer another response in the form of a theory about rational inferences from samples, one which places an emphasis on the role of intuition in interpreting the probabilistic implications of evidence. The theory is nevertheless consistent with formal methods of statistical analysis in many contexts (such as objective Bayesian analyses of random samples). In accordance with the theory, one may use evidence from samples to form probability distributions about the relevant relative frequencies in a population. I then sketch out how the resulting distributions can be integrated with the insights from the random-worlds method à la the theorem of total probability. This, then, provides an approach to constraining probabilities given evidence about relative frequencies.