Some observations on inverse probability including a new indifference rule
The main object of this paper is to propound and discuss a new indifference rule for the prior probabilities in the theory of inverse probability. Being invariant in form on transformation, this new rule avoids the mathematical inconsistencies associated with the classical rule of ‘uniform distribution of ignorance’ and yields results which, particularly in certain critical extreme cases, do not appear to be unreasonable. Such a rule is, of course, a postulate and is not susceptible of proof; its object is to enable inverse probability to operate as a unified principle upon which methods may be devised of allowing a set of statistics to tell their complete and unbiased story about the parameters of the distribution law of the population from which they have been drawn, without the introduction of any knowledge beyond and extraneous to the statistics themselves. The forms appropriate for the prior probabilities in certain other circumstances are also discussed, including the important case where the unknown parameter is a probability, or proportion, for which it is desired to allow for prior bias.