[1] One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision.
[4] Bayesian epistemology, on the other hand, works by formalizing concepts and problems, which are often vague in the traditional approach.
[7][8] Justification plays a central role in traditional epistemology while Bayesians have focused on the related notions of confirmation and disconfirmation through evidence.
[5][6] Certain traditional problems, like the topic of skepticism about our knowledge of the external world, are difficult to express in Bayesian terms.
[1] This approach tries to capture the idea of certainty:[4] we believe in all kinds of claims but we are more certain about some, like that the earth is round, than about others, like that Plato was the author of the First Alcibiades.
A degree of 0.5 corresponds to suspension of belief, meaning that the person has not yet made up their mind: they have no opinion either way and thus neither accept nor reject the claim.
Following Frank P. Ramsey, they are interpreted in terms of the willingness to bet money on a claim.
[6] The reason for this is that ascribing these extreme values would mean that one would be willing to bet anything, including one's life, even if the payoff was minimal.
[1] Another negative side-effect of such extreme credences is that they are permanently fixed and cannot be updated anymore upon acquiring new evidence.
According to Stephan Hartmann and Jan Sprenger, the axioms of probability can be expressed through the following two laws: (1)
The axioms of probability together with the principal principle determines the static or synchronic aspect of rationality: what an agent's beliefs should be like when only considering one moment.
[1] But rationality also involves a dynamic or diachronic aspect, which comes to play for changing one's credences upon being confronted with new evidence.
The original expression of this principle, referred to as Bayes' theorem, can be directly deduced from this formulation.
[6] The simple principle of conditionalization makes the assumption that our credence in the acquired evidence, i.e. its posterior probability, is 1, which is unrealistic.
For example, scientists sometimes need to discard previously accepted evidence upon making new discoveries, which would be impossible if the corresponding credence was 1.
[6] An alternative form of conditionalization, proposed by Richard Jeffrey, adjusts the formula to take the probability of the evidence into account:[13][14]
On the traditional interpretation, such a vulnerability reveals that the agent is irrational since they would willingly engage in behavior that is not in their best self-interest.
[6] One problem with this interpretation is that it assumes logical omniscience as a requirement for rationality, which is problematic especially in complicated diachronic cases.
An alternative interpretation uses Dutch books as "a kind of heuristic for determining when one's degrees of belief have the potential to be pragmatically self-defeating".
[6] This interpretation is compatible with holding a more realistic view of rationality in the face of human limitations.
[6] A well-known problem in confirmation theory is Carl Gustav Hempel's raven paradox.
[6] Bayesianism allows that seeing a green apple supports the raven-hypothesis while explaining our initial intuition otherwise.
[4][23] Intuitively, this measures how likely it is that the two beliefs are true at the same time, compared to how likely this would be if they were neutrally related to each other.
[6] In this way, it can be formally shown that witness reports that are probabilistically independent of each other provide more support than otherwise.
[6][1] In order to draw probabilistic inferences based on new evidence, it is necessary to already have a prior probability assigned to the proposition in question.
This problem is usually solved by assigning a probability to the proposition in question in order to learn from the new evidence through conditionalization.
[25] Subjective Bayesians hold that there are no or few constraints besides probabilistic coherence that determine how we assign the initial probabilities.
[6] Applied to this situation, the principle of indifference states that the agent should initially assume that the probability to draw a red ball is 50%.
This is due to symmetric considerations: it is the only assignment in which the prior probabilities are invariant to a change in label.
But this is not allowed in Bayesian confirmation theory since conditionalization can only happen upon a change of the probability of the evidential statement, which is not the case.