As repeatedly noted by me and by many others, there are multiple theories about the fundamental notion of probability, including (though not restricted to) the notion of probabilities as objective, logical relationships amongst propositions and that of probabilities as degrees of belief.
Though those two notions are distinct, subscribers to each typically agree with subscribers to the other upon a great deal of the axiomatic structure of the logic of probability. Further, in practice the main-stream of the first group and that of the second group both arrive at their estimates of measures of probability by adjusting initial values through repeated application, as observations accumulate, of a principle known as
Bayes' theorem. Indeed, the main-stream of one group are called
objective Bayesian and the mainstream of the other are often called
subjective Bayesian. Where the two main-streams differ in practice is in the source of those initial values.
The objective Bayesians believe that, in the absence of information, one begins with what are called
non-informative priors. This notion is evolved from the classical idea of a principle of insufficient reason, which said that one should assign equal probabilities to events or to propositions, in the absence of a reason for assigning different probabilities. (For example, begin by assume that a die is
fair.) The objective Bayesians attempt to be more shrewd than the classical theorists, but will often admit that in some cases non-informative priors cannot be found because of a lack of understanding of how to divide the possibilities (in some cases because of complexity).
The subjective Bayesians believe that one may use as a prior whatever initial degree of belief one has, measured on an interval from 0 through 1. As measures of probability are taken to be degrees of belief, any application of Bayes' theorem that results in a new value is supposed to result in a new degree of belief.
I want to suggest what I think to be a new school of thought, with a Bayesian sub-school, not-withstanding that I have no intention of joining this school.
If a set of things is completely ranked, it's possible to proxy that ranking with a quantification, such that if one thing has a higher rank than another then it is assigned a greater quantification, and that if two things have the same rank then they are assigned the same quantification. If all that we have is a ranking, with no further stipulations, then there will be infinitely many possible quantifications that will work as proxies. Often, we may want to tighten-up the rules of quantification (for example, by requiring that all quantities be in the interval from 0 through 1), and yet still it may be the case that infinitely many quantifications would work equally well as proxies.
Sets of measures of probability may be considered as proxies for underlying rankings of propositions or of events by probability. The principles to which most theorists agree when they consider probability rankings as such constrain the sets of possible measures, but so long as only a finite set of propositions or of events is under consideration, there are infinitely many sets of measures that will work as proxies.
A subjectivist feels free to use his or her degrees of belief so long as they fit the constraints, even though someone else may have a different set of degrees of belief that also fit the constraints. However, the argument for the admissibility of the subjectivist's own set of degrees of belief is not that it is believed; the argument is that one's own set of degrees of belief fits the constraints. Belief as such is irrelevant. It might be that one's own belief is colored by private information, but then the argument is not that one believes the private information, but that the information as such is relevant (as indeed it might be); and there would always be some other sets of measures that also conformed to the private information.
Perhaps one might as well use one's own set of degrees of belief, but one also might every bit as well use any conforming set of measures.
So what I now suggest is what I call a
libertine school, which regards measures of probability as proxies for probability rankings and which accepts any set of measures that conform to what is known of the probability ranking of propositions or of events, regardless of whether these measures are thought to be the degrees of belief of anyone, and without any concern that these should become the degrees of belief of anyone; and in particular I suggest
libertine Bayesianism, which accepts the analytic principles common to the objective Bayesians and to the subjective Bayesians, but which will allow any set of priors that conforms to those principles.
 So great a share of subjectivists subscribe to a Bayesian principle of updating that often the subjective Bayesians are simply called
subjectivists as if there were no need to distinguish amongst subjectivists. And, until relatively recently, so little recognition was given to the objective Bayesians that
Bayesian was often taken as synonymous with