It is popular to divide Bayesians into two main categories, “objective” and “subjective” Bayesians. The divide is sometimes made formal, there are conferences labelled as one but not the other, for example.
A caricature of subjective Bayes is that all probabilities are just opinion, and the best we can do with an opinion is make sure it isn’t self contradictory, and satisfying the rules of probability is a way of ensuring that. A caricature of objective Bayes is that there exists a correct probability for every hypothesis given certain information, and that different people with the same information should make exactly the same probability judgments.
My view is that the distinction between subjective and objective Bayes is not helpful, and that something in between is usually desirable. The purpose of Bayesian inference (and ME updating) is to study how an ideal perfect reasoner would respond to certain pieces of information. When I choose a hypothesis space and a prior distribution, I’m not writing down my prior beliefs. That would be impossible, since probabilities aren’t actually a thing that exist in my brain. I am writing down a hypothetical state of prior beliefs that may or may not be a good approximation to what I (or some expert) think. But whatever I use in a calculation is always an idealisation, just like all models in physics.
If I want the posterior distribution to capture pretty much all of the information available about a question (for example, because we need to make an important decision), then I will do things that resemble subjective Bayes, for example, deriving various consequences of the prior probability assignment and making sure I (or someone who knows more about the topic than me) agree with them. On the other hand, if I want to understand the process of learning from a certain kind of data, then I will use various types of flat priors in an attempt to keep things simple so I can see what is going on.
I would like to finish with this (long) quote from physicist Ariel Caticha, that says similar things but in a different way.
“Diﬀerent individuals may hold diﬀerent beliefs and it is certainly important to ﬁgure out what those beliefs might be — perhaps by observing their gambling behavior — but this is not our present concern. Our objective is neither to assess nor to describe the subjective beliefs of any particular individual. Instead we deal with the altogether diﬀerent but very common problem that arises when we are confused and we want some guidance about what we are supposed to believe. Our concern here is not so much with beliefs as they actually are, but rather, with beliefs as they ought to be. Rational beliefs are constrained beliefs. Indeed, the essence of rationality lies precisely in the existence of some constraints. The problem, of course, is to ﬁgure out what those constraints might be. We need to identify normative criteria of rationality. It must be stressed that the beliefs discussed here are meant to be those held by an idealized rational individual who is not subject to practical human limitations. We are concerned with those ideal standards of rationality that we ought to strive to attain at least when discussing scientiﬁc matters. Here is our ﬁrst criterion of rationality: whatever guidelines we pick they must be of general applicability—otherwise they fail when most needed, namely, when not much is known about a problem. Diﬀerent rational individuals can reason about diﬀerent topics, or about the same subject but on the basis of diﬀerent information, and therefore they could hold diﬀerent beliefs, but they must agree to follow the same rules.”