Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. It is based on the idea that beliefs can be interpreted as subjective probabilities. As such, they are subject to the laws of probability theory, which act as the norms of rationality. These norms can be divided into static constraints, governing the rationality of beliefs at any moment, and dynamic constraints, governing how rational agents should change their beliefs upon receiving new evidence. 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.
The most characteristic Bayesian expression of these principles is found in the form of Dutch books, which illustrate irrationality in agents through a series of bets that lead to a loss for the agent no matter which of the probabilistic events occurs. Bayesians have applied these fundamental principles to various epistemological topics but Bayesianism does not cover all topics of traditional epistemology. The problem of confirmation in the philosophy of science, for example, can be approached through the Bayesian principle of conditionalization by holding that a piece of evidence confirms a theory if it raises the likelihood that this theory is true. Various proposals have been made to define the concept of coherence in terms of probability, usually in the sense that two propositions cohere if the probability of their conjunction is higher than if they were neutrally related to each other. The Bayesian approach has also been fruitful in the field of social epistemology, for example, concerning the problem of testimony or the problem of group belief. Bayesianism still faces various theoretical objections that have not been fully solved.
Relation to traditional epistemology Traditional epistemology and Bayesian epistemology are both forms of epistemology, but they differ in various respects, for example, concerning their methodology, their interpretation of belief, the role justification or confirmation plays in them and some of their research interests. Traditional epistemology focuses on topics such as the analysis of the nature of knowledge, usually in terms of justified true beliefs, the sources of knowledge, like perception or testimony, the structure of a body of knowledge, for example in the form of foundationalism or coherentism, and the problem of philosophical skepticism or the question of whether knowledge is possible at all. These inquiries are usually based on epistemic intuitions and regard beliefs as either present or absent. Bayesian epistemology, on the other hand, works by formalizing concepts and problems, which are often vague in the traditional approach. It thereby focuses more on mathematical intuitions and promises a higher degree of precision. It sees belief as a continuous phenomenon that comes in various degrees, so-called credences. Some Bayesians have even suggested that the regular notion of belief should be abandoned. But there are also proposals to connect the two, for example, the Lockean thesis, which defines belief as credence above a certain threshold. Justification plays a central role in traditional epistemology while Bayesians have focused on the related notions of confirmation and disconfirmation through evidence. The notion of evidence is important for both approaches but only the traditional approach has been interested in studying the sources of evidence, like perception and memory. Bayesianism, on the other hand, has focused on the role of evidence for rationality: how someone's credence should be adjusted upon receiving new evidence. There is an analogy between the Bayesian norms of rationality in terms of probabilistic laws and the traditional norms of rationality in terms of deductive consistency. Certain traditional problems, like the topic of skepticism about our knowledge of the external world, are difficult to express in Bayesian terms.
Fundamentals Bayesian epistemology is based only on a few fundamental principles, which can be used to define various other notions and can be applied to many topics in epistemology. At their core, these principles constitute constraints on how we should assign credences to propositions. They determine what an ideally rational agent would believe. The basic principles can be divided into synchronic or static principles, which govern how credences are to be assigned at any moment, and diachronic or dynamic principles, which determine how the agent should change their beliefs upon receiving new evidence. The axioms of probability and the principal principle belong to the static principles while the principle of conditionalization governs the dynamic aspects as a form of probabilistic inference. The most characteristic Bayesian expression of these principles is found in the form of Dutch books, which illustrate irrationality in agents through a series of bets that lead to a loss for the agent no matter which of the probabilistic events occurs. This test for determining irrationality has been referred to as the "pragmatic self-defeat test".
Beliefs, probability, and bets One important difference to traditional epistemology is that Bayesian epistemology focuses not on the notion of simple belief but on the notion of degrees of belief, so-called credences. This approach tries to capture the idea of certainty: 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. These degrees come in values between 0 and 1. A degree of 1 implies that a claim is completely accepted. A degree of 0, on the other hand, corresponds to full disbelief. This means that the claim is fully rejected and the person firmly believes the opposite claim. 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. According to the Bayesian interpretation of probability, credences stand for subjective probabilities. Following Frank P. Ramsey, they are interpreted in terms of the willingness to bet money on a claim. So having a credence of 0.8 (i.e. 80 %) that your favorite soccer team will win the next game would mean being willing to bet up to four dollars for the chance to make one dollar profit. This account draws a tight connection between Bayesian epistemology and decision theory. It might seem that betting-behavior is only one special area and as such not suited for defining such a general notion as credences. But, as Ramsey argues, we bet all the time when understood in the widest sense. For example, in going to the train station, we bet on the train being there on time, otherwise we would have stayed at home. It follows from the interpretation of credence in terms of willingness to make bets that it would be irrational to ascribe a credence of 0 or 1 to any proposition, except for contradictions and tautologies. 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. Another negative side-effect of such extreme credences is that they are permanently fixed and cannot be updated anymore upon acquiring new evidence.
This central tenet of Bayesianism, that credences are interpreted as subjective probabilities and are therefore governed by the norms of probability, has been referred to as probabilism. These norms express the nature of the credences of ideally rational agents. They do not put demands on what credence we should have on any single given belief, for example, whether it will rain tomorrow. Instead, they constrain the system of beliefs as a whole. For example, if your credence that it will rain tomorrow is 0.8 then your credence in the opposite proposition, i.e. that it will not rain tomorrow, should be 0.2, not 0.1 or 0.5. According to Stephan Hartmann and Jan Sprenger, the axioms of probability can be expressed through the following two laws: (1) for any tautology ; (2) For incompatible (mutually exclusive) propositions and , .
Another important Bayesian principle of degrees of beliefs is the principal principle due to David Lewis. It states that our knowledge of objective probabilities should correspond to our subjective probabilities in the form of credences. So if you know that the objective chance of a coin landing heads is 50% then your credence that the coin will land heads should be 0.5.
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. But rationality also involves a dynamic or diachronic aspect, which comes to play for changing one's credences upon being confronted with new evidence. This aspect is determined by the principle of conditionalization.
Principle of conditionalization The principle of conditionalization governs how the agent's credence in a hypothesis should change upon receiving new evidence for or against this hypothesis. As such, it expresses the dynamic aspect of how ideal rational agents would behave. It is based on the notion of conditional probability, which is the measure of the probability that one event occurs given that another event has already occurred. The unconditional probability that will occur is usually expressed as while the conditional probability that will occur given that B has already occurred is written as . For example, the probability of flipping a coin two times and the coin landing heads two times is only 25%. But the conditional probability of this occurring given that the coin has landed heads on the first flip is then 50%. The principle of conditionalization applies this idea to credences: we should change our credence that the coin will land heads two times upon receiving evidence that it has already landed heads on the first flip. The probability assigned to the hypothesis before the event is called prior probability. The probability afterward is called posterior probability. According to the simple principle of conditionalization, this can be expressed in the following way: . So the posterior probability that the hypothesis is true is equal to the conditional prior probability that the hypothesis is true relative to the evidence, which is equal to the prior probability that both the hypothesis and the evidence are true, divided by the prior probability that the evidence is true. The original expression of this principle, referred to as Bayes' theorem, can be directly deduced from this formulation.
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. An alternative form of conditionalization, proposed by Richard Jeffrey, adjusts the formula to take the probability of the evidence into account: .
Dutch books