Bayesian confirmation theory tells you to update your credences by conditionalizing on any new evidence you encounter. However, conditionalization depends on your prior credences, and Bayesianism does not tell you what those prior credences should be. The only requirement is that your prior credences must be probabilistically coherent, as defined previously. Two people with different initial credences—different “starting points,” so to speak—could reach radically different conclusions when they conditionalize on the same evidence. One person could regard evidence E as strong support for hypothesis H; another could regard E as strong evidence against H, even though both individuals are following the conditionalization rule. Is there no objective fact of the matter whether E really confirms H? Is any initial credence as good as any other, or are some starting points more reasonable than others? The Bayesian theory of confirmation, by itself, gives no answer to these questions.
These unanswered questions have engendered an ongoing debate between two schools of thought within Bayesianism confirmation theory. Subjective Bayesians argue that there are no rational requirements for prior credences except probabilistic coherence (i.e., conformity to the rules of probability). Objective Bayesians, on the contrary, see the subjectivity of Bayesianism as a serious problem for the theory. In order to make it more objective, they have tried to identify rational requirements or rules for prior credences, in addition to the minimum requirement of probabilistic coherence. Many objective Bayesians appeal to the so-called principle of indifference, which says that when you have no evidence to decide between multiple hypotheses, you should assign the same probability to each of those hypotheses. Depending on how the hypotheses in question are enumerated, however, the principle of indifference yields contradictory results, and there is no consensus on how to resolve that issue.
My own opinion, for what it’s worth, lies somewhere between those two positions. On the one hand, I agree with objective Bayesians that mere probabilistic coherence is insufficient to make one’s starting point reasonable. However, I am not optimistic that they will succeed in finding strict rules of rationality for prior credences, beyond the rules of probabilistic coherence. As I see it, the difference between “reasonable” and “unreasonable” starting points is largely a matter of common sense, which cannot be formulated into a precise (much less concise) set of rules. So, I think the best approach is to employ Bayesian methodology in the broad range of cases where its subjectivity isn’t a problem: for example, when working out the implications of our own beliefs, and when engaging in philosophical debates on which there is some level of agreement on the relevant prior probabilities. When there is disagreement on what the priors should be, the inherent subjectivity of Bayesianism may render it less helpful.